P. 265 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26401-26500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cos2pi 26401	The cosine of 2π is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
⊢ (cos‘(2 · π)) = 1
Theorem	ef2pi 26402	The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
⊢ (exp‘(i · (2 · π))) = 1
Theorem	ef2kpi 26403	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 26404	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 26405	Lemma for sinper 26406 and cosper 26407. (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 26406	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 26407	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 26408	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 26409	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 26410	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 26411	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 26412	Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴))
Theorem	cosmpi 26413	Cosine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 − π)) = -(cos‘𝐴))
Theorem	sinppi 26414	Sine of a number plus π. (Contributed by NM, 10-Aug-2008.)
⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 + π)) = -(sin‘𝐴))
Theorem	cosppi 26415	Cosine of a number plus π. (Contributed by NM, 18-Aug-2008.)
⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴))
Theorem	efimpi 26416	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 26417	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 26418	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 26419	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 26420	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 26421	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 16099, 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 26422	Lemma for sincosq1sgn 26423. (Contributed by Paul Chapman, 24-Jan-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴))
Theorem	sincosq1sgn 26423	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 26424	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 26425	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 26426	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 26427	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 26428	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 26429	Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
⊢ (𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)
Theorem	tangtx 26430	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 26431	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 26432	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 26433	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 26434	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 26435	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 26436	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 26437	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 26438	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 26439	The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.) (Proof shortened by SN, 2-Sep-2025.)
⊢ (tan‘(π / 4)) = 1
Theorem	tan4thpiOLD 26440	Obsolete version of tan4thpi 26439 as of 2-Sep-2025. (Contributed by Mario Carneiro, 5-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (tan‘(π / 4)) = 1
Theorem	sincos6thpi 26441	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 26442	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 26443	π is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ 3 < π
Theorem	pige3 26444	π is greater than or equal to 3. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ 3 ≤ π
Theorem	pige3ALT 26445	Alternate proof of pige3 26444. 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 26446	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 26447	The sine of an integer multiple of π is 0. (Contributed by NM, 11-Aug-2008.)
⊢ (𝐾 ∈ ℤ → (sin‘(𝐾 · π)) = 0)
Theorem	coskpi 26448	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 26449	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 26450	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 26451	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 26452	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 26453	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 26454	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 26455	Lemma for cosord 26456. (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 26456	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 26457	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 26458	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 26459	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 26460	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)
Theorem	resinf1o 26461	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)
Theorem	tanord1 26462	The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 26463.) (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)))
Theorem	tanord 26463	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)))
Theorem	tanregt0 26464	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)))
Theorem	negpitopissre 26465	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
Theorem	efgh 26466*	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 = (𝑥 ∈ 𝑋 ↦ (exp‘(𝐴 · 𝑥)))    ⇒   ⊢ (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐹‘(𝐵 + 𝐶)) = ((𝐹‘𝐵) · (𝐹‘𝐶)))
Theorem	efif1olem1 26467*	Lemma for efif1o 26471. (Contributed by Mario Carneiro, 13-May-2014.)
⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π)))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))
Theorem	efif1olem2 26468*	Lemma for efif1o 26471. (Contributed by Mario Carneiro, 13-May-2014.)
⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π)))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ)
Theorem	efif1olem3 26469*	Lemma for efif1o 26471. (Contributed by Mario Carneiro, 8-May-2015.)
⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))    &   ⊢ 𝐶 = (◡abs “ {1})    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (ℑ‘(√‘𝑥)) ∈ (-1[,]1))
Theorem	efif1olem4 26470*	The exponential function of an imaginary number maps any interval of length 2π one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)
⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))    &   ⊢ 𝐶 = (◡abs “ {1})    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))    &   ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ)    &   ⊢ 𝑆 = (sin ↾ (-(π / 2)[,](π / 2)))    ⇒   ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝐶)
Theorem	efif1o 26471*	The exponential function of an imaginary number maps any open-below, closed-above interval of length 2π one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.)
⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))    &   ⊢ 𝐶 = (◡abs “ {1})    &   ⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π)))    ⇒   ⊢ (𝐴 ∈ ℝ → 𝐹:𝐷–1-1-onto→𝐶)
Theorem	efifo 26472*	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–onto→𝐶
Theorem	eff1olem 26473*	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 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))    &   ⊢ 𝑆 = (◡ℑ “ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))    &   ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ)    ⇒   ⊢ (𝜑 → (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}))
Theorem	eff1o 26474	The exponential function maps the set 𝑆, of complex numbers with imaginary part in the closed-above, open-below interval from -π 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):𝑆–1-1-onto→(ℂ ∖ {0})
Theorem	efabl 26475*	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 = (𝑥 ∈ 𝑋 ↦ (exp‘(𝐴 · 𝑥)))    &   ⊢ 𝐺 = ((mulGrp‘ℂfld) ↾s ran 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘ℂfld))    ⇒   ⊢ (𝜑 → 𝐺 ∈ Abel)
Theorem	efsubm 26476*	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 ℂfld. (Contributed by Thierry Arnoux, 26-Jan-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (exp‘(𝐴 · 𝑥)))    &   ⊢ 𝐺 = ((mulGrp‘ℂfld) ↾s ran 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘ℂfld))    ⇒   ⊢ (𝜑 → ran 𝐹 ∈ (SubMnd‘(mulGrp‘ℂfld)))
Theorem	circgrp 26477	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
Theorem	circsubm 26478	The circle group 𝑇 is a submonoid of the multiplicative group of ℂfld. (Contributed by Thierry Arnoux, 26-Jan-2020.)
⊢ 𝐶 = (◡abs “ {1})    &   ⊢ 𝑇 = ((mulGrp‘ℂfld) ↾s 𝐶)    ⇒   ⊢ 𝐶 ∈ (SubMnd‘(mulGrp‘ℂfld))
14.3.4  The natural logarithm on complex numbers
Syntax	clog 26479	Extend class notation with the natural logarithm function on complex numbers.
class log
Syntax	ccxp 26480	Extend class notation with the complex power function.
class ↑𝑐
Definition	df-log 26481	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)))
Definition	df-cxp 26482*	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)))))
Theorem	logrn 26483	The range of the natural logarithm function, also the principal domain of the exponential function. This allows 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))
Theorem	ellogrn 26484	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))
Theorem	dflog2 26485	The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008.)
⊢ log = ◡(exp ↾ ran log)
Theorem	relogrn 26486	The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 1-Apr-2015.)
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ran log)
Theorem	logrncn 26487	The range of the natural logarithm function is a subset of the complex numbers. (Contributed by Mario Carneiro, 13-May-2014.)
⊢ (𝐴 ∈ ran log → 𝐴 ∈ ℂ)
Theorem	eff1o2 26488	The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
⊢ (exp ↾ ran log):ran log–1-1-onto→(ℂ ∖ {0})
Theorem	logf1o 26489	The natural logarithm function maps the nonzero complex numbers one-to-one onto its range. (Contributed by Paul Chapman, 21-Apr-2008.)
⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
Theorem	dfrelog 26490	The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
⊢ (log ↾ ℝ+) = ◡(exp ↾ ℝ)
Theorem	relogf1o 26491	The natural logarithm function maps the positive reals one-to-one onto the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
⊢ (log ↾ ℝ+):ℝ+–1-1-onto→ℝ
Theorem	logrncl 26492	Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ran log)
Theorem	logcl 26493	Closure of the natural logarithm function. (Contributed by NM, 21-Apr-2008.) (Revised by Mario Carneiro, 23-Sep-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
Theorem	logimcl 26494	Closure of the imaginary part of the logarithm function. (Contributed by Mario Carneiro, 23-Sep-2014.) (Revised by Mario Carneiro, 1-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
Theorem	logcld 26495	The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 26493. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (log‘𝑋) ∈ ℂ)
Theorem	logimcld 26496	The imaginary part of the logarithm is in (-π(,]π). Deduction form of logimcl 26494. Compare logimclad 26497. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (-π < (ℑ‘(log‘𝑋)) ∧ (ℑ‘(log‘𝑋)) ≤ π))
Theorem	logimclad 26497	The imaginary part of the logarithm is in (-π(,]π). Alternate form of logimcld 26496. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (ℑ‘(log‘𝑋)) ∈ (-π(,]π))
Theorem	abslogimle 26498	The imaginary part of the logarithm function has absolute value less than pi. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
Theorem	logrnaddcl 26499	The range of the natural logarithm is closed under addition with reals. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ran log)
Theorem	relogcl 26500	Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)