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Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabelth2 24701* 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 24702 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
exp ∈ (ℂ–cn→ℂ)

Theoremsincn 24703 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
sin ∈ (ℂ–cn→ℂ)

Theoremcoscn 24704 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
cos ∈ (ℂ–cn→ℂ)

Theoremreeff1olem 24705* Lemma for reeff1o 24706. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)

Theoremreeff1o 24706 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 24707 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 24708 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 24709 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 24710 Lemma for pire 24715, pigt2lt4 24713 and sinpi 24714. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐴 ∈ (ℝ+ ∩ (sin “ {0})) ↔ (𝐴 ∈ ℝ+ ∧ (sin‘𝐴) = 0))

Theorempilem2 24711 Lemma for pire 24715, pigt2lt4 24713 and sinpi 24714. (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 24712 Lemma for pire 24715, pigt2lt4 24713 and sinpi 24714. 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 24713 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 24714 The sine of π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘π) = 0

Theorempire 24715 π is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
π ∈ ℝ

Theorempicn 24716 π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
π ∈ ℂ

Theorempipos 24717 π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
0 < π

Theorempirp 24718 π is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
π ∈ ℝ+

Theoremnegpicn 24719 is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
-π ∈ ℂ

Theoremsinhalfpilem 24720 Lemma for sinhalfpi 24725 and coshalfpi 24726. (Contributed by Paul Chapman, 23-Jan-2008.)
((sin‘(π / 2)) = 1 ∧ (cos‘(π / 2)) = 0)

Theoremhalfpire 24721 π / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
(π / 2) ∈ ℝ

Theoremneghalfpire 24722 -π / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ

Theoremneghalfpirx 24723 -π / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ*

Theorempidiv2halves 24724 Adding π / 2 to itself gives π. See 2halves 11702. (Contributed by David A. Wheeler, 8-Dec-2018.)
((π / 2) + (π / 2)) = π

Theoremsinhalfpi 24725 The sine of π / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(π / 2)) = 1

Theoremcoshalfpi 24726 The cosine of π / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(π / 2)) = 0

Theoremcosneghalfpi 24727 The cosine of -π / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
(cos‘-(π / 2)) = 0

Theoremefhalfpi 24728 The exponential of iπ / 2 is i. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (π / 2))) = i

Theoremcospi 24729 The cosine of π is -1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘π) = -1

Theoremefipi 24730 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 24731 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
((exp‘(i · π)) + 1) = 0

Theoremsin2pi 24732 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(2 · π)) = 0

Theoremcos2pi 24733 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(2 · π)) = 1

Theoremef2pi 24734 The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (2 · π))) = 1

Theoremef2kpi 24735 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 24736 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 24737 Lemma for sinper 24738 and cosper 24739. (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 24738 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 24739 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 24740 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 24741 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 24742 Sine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘((2 · π) − 𝐴)) = -(sin‘𝐴))

Theoremcos2pim 24743 Cosine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘((2 · π) − 𝐴)) = (cos‘𝐴))

Theoremsinmpi 24744 Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴))

Theoremcosmpi 24745 Cosine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 − π)) = -(cos‘𝐴))

Theoremsinppi 24746 Sine of a number plus π. (Contributed by NM, 10-Aug-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 + π)) = -(sin‘𝐴))

Theoremcosppi 24747 Cosine of a number plus π. (Contributed by NM, 18-Aug-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴))

Theoremefimpi 24748 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)))

Theoremsinhalfpip 24749 The sine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))

Theoremsinhalfpim 24750 The sine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))

Theoremcoshalfpip 24751 The cosine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) + 𝐴)) = -(sin‘𝐴))

Theoremcoshalfpim 24752 The cosine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) − 𝐴)) = (sin‘𝐴))

Theoremptolemy 24753 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 15346, 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 24754 Lemma for sincosq1sgn 24755. (Contributed by Paul Chapman, 24-Jan-2008.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴𝐴 < (π / 2)) → 0 < (sin‘𝐴))

Theoremsincosq1sgn 24755 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 24756 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 24757 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 24758 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 24759 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 24760 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 24761 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)

Theoremtangtx 24762 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 24763 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 24764 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 24765 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 24766 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 24767 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))

Theoremcosq14ge0 24768 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))

Theoremsincosq1eq 24769 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))))

Theoremsincos4thpi 24770 The sine and cosine of π / 4. (Contributed by Paul Chapman, 25-Jan-2008.)
((sin‘(π / 4)) = (1 / (√‘2)) ∧ (cos‘(π / 4)) = (1 / (√‘2)))

Theoremtan4thpi 24771 The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.)
(tan‘(π / 4)) = 1

Theoremsincos6thpi 24772 The sine and cosine of π / 6. (Contributed by Paul Chapman, 25-Jan-2008.) Replace OLD theorem. (Revised by Wolf Lammen, 24-Sep-2020.)
((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))

Theoremsincos3rdpi 24773 The sine and cosine of π / 3. (Contributed by Mario Carneiro, 21-May-2016.)
((sin‘(π / 3)) = ((√‘3) / 2) ∧ (cos‘(π / 3)) = (1 / 2))

Theorempigt3 24774 π is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)
3 < π

Theorempige3 24775 π is greater than or equal to 3. (Contributed by Mario Carneiro, 21-May-2016.)
3 ≤ π

Theorempige3ALT 24776 Alternate proof of pige3 24775. 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 . 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

Theoremabssinper 24777 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)))

Theoremsinkpi 24778 The sine of an integer multiple of π is 0. (Contributed by NM, 11-Aug-2008.)
(𝐾 ∈ ℤ → (sin‘(𝐾 · π)) = 0)

Theoremcoskpi 24779 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)

Theoremsineq0 24780 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))

Theoremcoseq1 24781 A complex number whose cosine is one is an integer multiple of . (Contributed by Mario Carneiro, 12-May-2014.)
(𝐴 ∈ ℂ → ((cos‘𝐴) = 1 ↔ (𝐴 / (2 · π)) ∈ ℤ))

Theoremefeq1 24782 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))

Theoremcosne0 24783 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)

Theoremcosordlem 24784 Lemma for cosord 24785. (Contributed by Mario Carneiro, 10-May-2014.)
(𝜑𝐴 ∈ (0[,]π))    &   (𝜑𝐵 ∈ (0[,]π))    &   (𝜑𝐴 < 𝐵)       (𝜑 → (cos‘𝐵) < (cos‘𝐴))

Theoremcosord 24785 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)))

Theoremcos11 24786 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)))

Theoremsinord 24787 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)))

Theoremrecosf1o 24788 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)

Theoremresinf1o 24789 The sine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)
(sin ↾ (-(π / 2)[,](π / 2))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1)

Theoremtanord1 24790 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 24791.) (Contributed by Mario Carneiro, 29-Jul-2014.) Revised to replace an OLD theorem. (Revised by Wolf Lammen, 20-Sep-2020.)
((𝐴 ∈ (0[,)(π / 2)) ∧ 𝐵 ∈ (0[,)(π / 2))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))

Theoremtanord 24791 The tangent function is strictly increasing on its principal domain. (Contributed by Mario Carneiro, 4-Apr-2015.)
((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))

Theoremtanregt0 24792 The real part of the tangent of a complex number with real part in the open interval (0(,)(π / 2)) is positive. (Contributed by Mario Carneiro, 5-Apr-2015.)
((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (0(,)(π / 2))) → 0 < (ℜ‘(tan‘𝐴)))

Theoremnegpitopissre 24793 The interval (-π(,]π) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
(-π(,]π) ⊆ ℝ

14.3.3  Mapping of the exponential function

Theoremefgh 24794* The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 11-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
𝐹 = (𝑥𝑋 ↦ (exp‘(𝐴 · 𝑥)))       (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵𝑋𝐶𝑋) → (𝐹‘(𝐵 + 𝐶)) = ((𝐹𝐵) · (𝐹𝐶)))

Theoremefif1olem1 24795* Lemma for efif1o 24799. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ (𝑥𝐷𝑦𝐷)) → (abs‘(𝑥𝑦)) < (2 · π))

Theoremefif1olem2 24796* Lemma for efif1o 24799. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ∃𝑦𝐷 ((𝑧𝑦) / (2 · π)) ∈ ℤ)

Theoremefif1olem3 24797* Lemma for efif1o 24799. (Contributed by Mario Carneiro, 8-May-2015.)
𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))    &   𝐶 = (abs “ {1})       ((𝜑𝑥𝐶) → (ℑ‘(√‘𝑥)) ∈ (-1[,]1))

Theoremefif1olem4 24798* The exponential function of an imaginary number maps any interval of length one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)
𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))    &   𝐶 = (abs “ {1})    &   (𝜑𝐷 ⊆ ℝ)    &   ((𝜑 ∧ (𝑥𝐷𝑦𝐷)) → (abs‘(𝑥𝑦)) < (2 · π))    &   ((𝜑𝑧 ∈ ℝ) → ∃𝑦𝐷 ((𝑧𝑦) / (2 · π)) ∈ ℤ)    &   𝑆 = (sin ↾ (-(π / 2)[,](π / 2)))       (𝜑𝐹:𝐷1-1-onto𝐶)

Theoremefif1o 24799* The exponential function of an imaginary number maps any open-below, closed-above interval of length one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.)
𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))    &   𝐶 = (abs “ {1})    &   𝐷 = (𝐴(,](𝐴 + (2 · π)))       (𝐴 ∈ ℝ → 𝐹:𝐷1-1-onto𝐶)

Theoremefifo 24800* The exponential function of an imaginary number maps the reals onto the unit circle. (Contributed by Mario Carneiro, 13-May-2014.)
𝐹 = (𝑧 ∈ ℝ ↦ (exp‘(i · 𝑧)))    &   𝐶 = (abs “ {1})       𝐹:ℝ–onto𝐶

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