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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
14.3  Basic trigonometry
 
14.3.1  The exponential, sine, and cosine functions (cont.)
 
Theoremefcn 24001 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
exp ∈ (ℂ–cn→ℂ)
 
Theoremsincn 24002 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
sin ∈ (ℂ–cn→ℂ)
 
Theoremcoscn 24003 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
cos ∈ (ℂ–cn→ℂ)
 
Theoremreeff1olem 24004* Lemma for reeff1o 24005. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
((𝑈 ∈ ℝ ∧ 1 < 𝑈) → ∃𝑥 ∈ ℝ (exp‘𝑥) = 𝑈)
 
Theoremreeff1o 24005 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 24006 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 24007 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 24008 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 24009 Lemma for pire 24014, pigt2lt4 24012 and sinpi 24013. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐴 ∈ (ℝ+ ∩ (sin “ {0})) ↔ (𝐴 ∈ ℝ+ ∧ (sin‘𝐴) = 0))
 
Theorempilem2 24010 Lemma for pire 24014, pigt2lt4 24012 and sinpi 24013. (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 24011 Lemma for pire 24014, pigt2lt4 24012 and sinpi 24013. 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 24012 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 24013 The sine of π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘π) = 0
 
Theorempire 24014 π is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
π ∈ ℝ
 
Theorempicn 24015 π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
π ∈ ℂ
 
Theorempipos 24016 π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
0 < π
 
Theorempirp 24017 π is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
π ∈ ℝ+
 
Theoremnegpicn 24018 is a real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-π ∈ ℂ
 
Theoremsinhalfpilem 24019 Lemma for sinhalfpi 24024 and coshalfpi 24025. (Contributed by Paul Chapman, 23-Jan-2008.)
((sin‘(π / 2)) = 1 ∧ (cos‘(π / 2)) = 0)
 
Theoremhalfpire 24020 π / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
(π / 2) ∈ ℝ
 
Theoremneghalfpire 24021 -π / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ
 
Theoremneghalfpirx 24022 -π / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ*
 
Theorempidiv2halves 24023 Adding π / 2 to itself is π (common case). See 2halves 11137. (Contributed by David A. Wheeler, 8-Dec-2018.)
((π / 2) + (π / 2)) = π
 
Theoremsinhalfpi 24024 The sine of π / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(π / 2)) = 1
 
Theoremcoshalfpi 24025 The cosine of π / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(π / 2)) = 0
 
Theoremcosneghalfpi 24026 The cosine of -π / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
(cos‘-(π / 2)) = 0
 
Theoremefhalfpi 24027 The exponential of iπ / 2 is i. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (π / 2))) = i
 
Theoremcospi 24028 The cosine of π is -1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘π) = -1
 
Theoremefipi 24029 The exponential of i · π. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
(exp‘(i · π)) = -1
 
Theoremeulerid 24030 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
((exp‘(i · π)) + 1) = 0
 
Theoremsin2pi 24031 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(2 · π)) = 0
 
Theoremcos2pi 24032 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(2 · π)) = 1
 
Theoremef2pi 24033 The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (2 · π))) = 1
 
Theoremef2kpi 24034 The exponential of 2𝐾πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐾 ∈ ℤ → (exp‘((i · (2 · π)) · 𝐾)) = 1)
 
Theoremefper 24035 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 24036 Lemma for sinper 24037 and cosper 24038. (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 24037 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 24038 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 24039 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 24040 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 24041 Sine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘((2 · π) − 𝐴)) = -(sin‘𝐴))
 
Theoremcos2pim 24042 Cosine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘((2 · π) − 𝐴)) = (cos‘𝐴))
 
Theoremsinmpi 24043 Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴))
 
Theoremcosmpi 24044 Cosine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 − π)) = -(cos‘𝐴))
 
Theoremsinppi 24045 Sine of a number plus π. (Contributed by NM, 10-Aug-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 + π)) = -(sin‘𝐴))
 
Theoremcosppi 24046 Cosine of a complex number plus π. (Contributed by NM, 18-Aug-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴))
 
Theoremefimpi 24047 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 24048 The sine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))
 
Theoremsinhalfpim 24049 The sine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
 
Theoremcoshalfpip 24050 The cosine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) + 𝐴)) = -(sin‘𝐴))
 
Theoremcoshalfpim 24051 The cosine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) − 𝐴)) = (sin‘𝐴))
 
Theoremptolemy 24052 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 14741, 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 24053 Lemma for sincosq1sgn 24054. (Contributed by Paul Chapman, 24-Jan-2008.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴𝐴 < (π / 2)) → 0 < (sin‘𝐴))
 
Theoremsincosq1sgn 24054 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 24055 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 24056 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 24057 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 24058 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 24059 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 24060 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)
 
Theoremtangtx 24061 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 24062 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 24063 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 24064 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 24065 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 24066 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 24067 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 24068 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 24069 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 24070 The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.)
(tan‘(π / 4)) = 1
 
Theoremsincos6thpi 24071 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 24072 The sine and cosine of π / 3. (Contributed by Mario Carneiro, 21-May-2016.)
((sin‘(π / 3)) = ((√‘3) / 2) ∧ (cos‘(π / 3)) = (1 / 2))
 
Theorempige3 24073 π is greater or equal to 3. 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.)
3 ≤ π
 
Theoremabssinper 24074 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 24075 The sine of an integer multiple of π is 0. (Contributed by NM, 11-Aug-2008.)
(𝐾 ∈ ℤ → (sin‘(𝐾 · π)) = 0)
 
Theoremcoskpi 24076 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 24077 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 24078 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 24079 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 24080 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 24081 Lemma for cosord 24082. (Contributed by Mario Carneiro, 10-May-2014.)
(𝜑𝐴 ∈ (0[,]π))    &   (𝜑𝐵 ∈ (0[,]π))    &   (𝜑𝐴 < 𝐵)       (𝜑 → (cos‘𝐵) < (cos‘𝐴))
 
Theoremcosord 24082 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 24083 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 24084 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 24085 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 24086 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 24087 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 24088.) (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 24088 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 24089 The positivity of tan(𝐴) extends to complex numbers with the same real part. (Contributed by Mario Carneiro, 5-Apr-2015.)
((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (0(,)(π / 2))) → 0 < (ℜ‘(tan‘𝐴)))
 
Theoremnegpitopissre 24090 (-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 24091* 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 24092* Lemma for efif1o 24096. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ (𝑥𝐷𝑦𝐷)) → (abs‘(𝑥𝑦)) < (2 · π))
 
Theoremefif1olem2 24093* Lemma for efif1o 24096. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ∃𝑦𝐷 ((𝑧𝑦) / (2 · π)) ∈ ℤ)
 
Theoremefif1olem3 24094* Lemma for efif1o 24096. (Contributed by Mario Carneiro, 8-May-2015.)
𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))    &   𝐶 = (abs “ {1})       ((𝜑𝑥𝐶) → (ℑ‘(√‘𝑥)) ∈ (-1[,]1))
 
Theoremefif1olem4 24095* 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 24096* 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 24097* 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
 
Theoremeff1olem 24098* The exponential function maps the set 𝑆, of complex numbers with imaginary part in a real interval of length 2 · π, one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)
𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))    &   𝑆 = (ℑ “ 𝐷)    &   (𝜑𝐷 ⊆ ℝ)    &   ((𝜑 ∧ (𝑥𝐷𝑦𝐷)) → (abs‘(𝑥𝑦)) < (2 · π))    &   ((𝜑𝑧 ∈ ℝ) → ∃𝑦𝐷 ((𝑧𝑦) / (2 · π)) ∈ ℤ)       (𝜑 → (exp ↾ 𝑆):𝑆1-1-onto→(ℂ ∖ {0}))
 
Theoremeff1o 24099 The exponential function maps the set 𝑆, of complex numbers with imaginary part in the closed-above, open-below interval from to π one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
𝑆 = (ℑ “ (-π(,]π))       (exp ↾ 𝑆):𝑆1-1-onto→(ℂ ∖ {0})
 
Theoremefabl 24100* The image of a subgroup of the group +, under the exponential function of a scaled complex number, is an Abelian group. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
𝐹 = (𝑥𝑋 ↦ (exp‘(𝐴 · 𝑥)))    &   𝐺 = ((mulGrp‘ℂfld) ↾s ran 𝐹)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝑋 ∈ (SubGrp‘ℂfld))       (𝜑𝐺 ∈ Abel)
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