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Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempilem3 24201 Lemma for pire 24204, pigt2lt4 24202 and sinpi 24203. 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 24202 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 24203 The sine of π is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘π) = 0

Theorempire 24204 π is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
π ∈ ℝ

Theorempicn 24205 π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
π ∈ ℂ

Theorempipos 24206 π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
0 < π

Theorempirp 24207 π is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
π ∈ ℝ+

Theoremnegpicn 24208 is a real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-π ∈ ℂ

Theoremsinhalfpilem 24209 Lemma for sinhalfpi 24214 and coshalfpi 24215. (Contributed by Paul Chapman, 23-Jan-2008.)
((sin‘(π / 2)) = 1 ∧ (cos‘(π / 2)) = 0)

Theoremhalfpire 24210 π / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
(π / 2) ∈ ℝ

Theoremneghalfpire 24211 -π / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ

Theoremneghalfpirx 24212 -π / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ*

Theorempidiv2halves 24213 Adding π / 2 to itself is π (common case). See 2halves 11257. (Contributed by David A. Wheeler, 8-Dec-2018.)
((π / 2) + (π / 2)) = π

Theoremsinhalfpi 24214 The sine of π / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(π / 2)) = 1

Theoremcoshalfpi 24215 The cosine of π / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(π / 2)) = 0

Theoremcosneghalfpi 24216 The cosine of -π / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
(cos‘-(π / 2)) = 0

Theoremefhalfpi 24217 The exponential of iπ / 2 is i. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (π / 2))) = i

Theoremcospi 24218 The cosine of π is -1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘π) = -1

Theoremefipi 24219 The exponential of i · π. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
(exp‘(i · π)) = -1

Theoremeulerid 24220 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
((exp‘(i · π)) + 1) = 0

Theoremsin2pi 24221 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(2 · π)) = 0

Theoremcos2pi 24222 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(2 · π)) = 1

Theoremef2pi 24223 The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (2 · π))) = 1

Theoremef2kpi 24224 The exponential of 2𝐾πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐾 ∈ ℤ → (exp‘((i · (2 · π)) · 𝐾)) = 1)

Theoremefper 24225 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 24226 Lemma for sinper 24227 and cosper 24228. (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 24227 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 24228 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 24229 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 24230 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 24231 Sine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘((2 · π) − 𝐴)) = -(sin‘𝐴))

Theoremcos2pim 24232 Cosine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘((2 · π) − 𝐴)) = (cos‘𝐴))

Theoremsinmpi 24233 Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴))

Theoremcosmpi 24234 Cosine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 − π)) = -(cos‘𝐴))

Theoremsinppi 24235 Sine of a number plus π. (Contributed by NM, 10-Aug-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 + π)) = -(sin‘𝐴))

Theoremcosppi 24236 Cosine of a complex number plus π. (Contributed by NM, 18-Aug-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴))

Theoremefimpi 24237 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 24238 The sine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))

Theoremsinhalfpim 24239 The sine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))

Theoremcoshalfpip 24240 The cosine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) + 𝐴)) = -(sin‘𝐴))

Theoremcoshalfpim 24241 The cosine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) − 𝐴)) = (sin‘𝐴))

Theoremptolemy 24242 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 14896, 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 24243 Lemma for sincosq1sgn 24244. (Contributed by Paul Chapman, 24-Jan-2008.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴𝐴 < (π / 2)) → 0 < (sin‘𝐴))

Theoremsincosq1sgn 24244 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 24245 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 24246 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 24247 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 24248 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 24249 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 24250 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)

Theoremtangtx 24251 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 24252 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 24253 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 24254 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 24255 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 24256 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 24257 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 24258 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 24259 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 24260 The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.)
(tan‘(π / 4)) = 1

Theoremsincos6thpi 24261 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 24262 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 24263 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 24264 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 24265 The sine of an integer multiple of π is 0. (Contributed by NM, 11-Aug-2008.)
(𝐾 ∈ ℤ → (sin‘(𝐾 · π)) = 0)

Theoremcoskpi 24266 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 24267 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 24268 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 24269 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 24270 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 24271 Lemma for cosord 24272. (Contributed by Mario Carneiro, 10-May-2014.)
(𝜑𝐴 ∈ (0[,]π))    &   (𝜑𝐵 ∈ (0[,]π))    &   (𝜑𝐴 < 𝐵)       (𝜑 → (cos‘𝐵) < (cos‘𝐴))

Theoremcosord 24272 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 24273 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 24274 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 24275 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 24276 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 24277 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 24278.) (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 24278 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 24279 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 24280 (-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 24281* 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 24282* Lemma for efif1o 24286. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ (𝑥𝐷𝑦𝐷)) → (abs‘(𝑥𝑦)) < (2 · π))

Theoremefif1olem2 24283* Lemma for efif1o 24286. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ∃𝑦𝐷 ((𝑧𝑦) / (2 · π)) ∈ ℤ)

Theoremefif1olem3 24284* Lemma for efif1o 24286. (Contributed by Mario Carneiro, 8-May-2015.)
𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))    &   𝐶 = (abs “ {1})       ((𝜑𝑥𝐶) → (ℑ‘(√‘𝑥)) ∈ (-1[,]1))

Theoremefif1olem4 24285* 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 24286* 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 24287* 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 24288* 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 24289 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 24290* 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)

Theoremefsubm 24291* The image of a subgroup of the group +, under the exponential function of a scaled complex number is a submonoid of the multiplicative group of fld. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝐹 = (𝑥𝑋 ↦ (exp‘(𝐴 · 𝑥)))    &   𝐺 = ((mulGrp‘ℂfld) ↾s ran 𝐹)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝑋 ∈ (SubGrp‘ℂfld))       (𝜑 → ran 𝐹 ∈ (SubMnd‘(mulGrp‘ℂfld)))

Theoremcircgrp 24292 The circle group 𝑇 is an Abelian group. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
𝐶 = (abs “ {1})    &   𝑇 = ((mulGrp‘ℂfld) ↾s 𝐶)       𝑇 ∈ Abel

Theoremcircsubm 24293 The circle group 𝑇 is a submonoid of the multiplicative group of fld. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝐶 = (abs “ {1})    &   𝑇 = ((mulGrp‘ℂfld) ↾s 𝐶)       𝐶 ∈ (SubMnd‘(mulGrp‘ℂfld))

Theoremrzgrp 24294 The quotient group R/Z is a group. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝑅 = (ℝfld /s (ℝfld ~QG ℤ))       𝑅 ∈ Grp

14.3.4  The natural logarithm on complex numbers

Syntaxclog 24295 Extend class notation with the natural logarithm function on complex numbers.
class log

Syntaxccxp 24296 Extend class notation with the complex power function.
class 𝑐

Definitiondf-log 24297 Define the natural logarithm function on complex numbers. It is defined as the principal value, that is, the inverse of the exponential whose imaginary part lies in the interval (-pi, pi]. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Paul Chapman, 21-Apr-2008.)
log = (exp ↾ (ℑ “ (-π(,]π)))

Definitiondf-cxp 24298* Define the power function on complex numbers. Note that the value of this function when 𝑥 = 0 and (ℜ‘𝑦) ≤ 0, 𝑦 ≠ 0 should properly be undefined, but defining it by convention this way simplifies the domain. (Contributed by Mario Carneiro, 2-Aug-2014.)
𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))

Theoremlogrn 24299 The range of the natural logarithm function, also the principal domain of the exponential function. This allows us to write the longer class expression as simply ran log. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
ran log = (ℑ “ (-π(,]π))

Theoremellogrn 24300 Write out the property 𝐴 ∈ ran log explicitly. (Contributed by Mario Carneiro, 1-Apr-2015.)
(𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))

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