 Home Metamath Proof ExplorerTheorem List (p. 240 of 425) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26947) Hilbert Space Explorer (26948-28472) Users' Mathboxes (28473-42426)

Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempire 23901 π is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
π ∈ ℝ

Theorempicn 23902 π is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
π ∈ ℂ

Theorempipos 23903 π is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
0 < π

Theorempirp 23904 π is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
π ∈ ℝ+

Theoremnegpicn 23905 is a real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-π ∈ ℂ

Theoremsinhalfpilem 23906 Lemma for sinhalfpi 23911 and coshalfpi 23912. (Contributed by Paul Chapman, 23-Jan-2008.)
((sin‘(π / 2)) = 1 ∧ (cos‘(π / 2)) = 0)

Theoremhalfpire 23907 π / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
(π / 2) ∈ ℝ

Theoremneghalfpire 23908 -π / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ

Theoremneghalfpirx 23909 -π / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
-(π / 2) ∈ ℝ*

Theorempidiv2halves 23910 Adding π / 2 to itself is π (common case). See 2halves 11015. (Contributed by David A. Wheeler, 8-Dec-2018.)
((π / 2) + (π / 2)) = π

Theoremsinhalfpi 23911 The sine of π / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(π / 2)) = 1

Theoremcoshalfpi 23912 The cosine of π / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(π / 2)) = 0

Theoremcosneghalfpi 23913 The cosine of -π / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
(cos‘-(π / 2)) = 0

Theoremefhalfpi 23914 The exponential of iπ / 2 is i. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (π / 2))) = i

Theoremcospi 23915 The cosine of π is -1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘π) = -1

Theoremefipi 23916 The exponential of i · π. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
(exp‘(i · π)) = -1

Theoremeulerid 23917 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
((exp‘(i · π)) + 1) = 0

Theoremsin2pi 23918 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
(sin‘(2 · π)) = 0

Theoremcos2pi 23919 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
(cos‘(2 · π)) = 1

Theoremef2pi 23920 The exponential of 2πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(exp‘(i · (2 · π))) = 1

Theoremef2kpi 23921 The exponential of 2𝐾πi is 1. (Contributed by Mario Carneiro, 9-May-2014.)
(𝐾 ∈ ℤ → (exp‘((i · (2 · π)) · 𝐾)) = 1)

Theoremefper 23922 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 23923 Lemma for sinper 23924 and cosper 23925. (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 23924 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 23925 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 23926 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 23927 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 23928 Sine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘((2 · π) − 𝐴)) = -(sin‘𝐴))

Theoremcos2pim 23929 Cosine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘((2 · π) − 𝐴)) = (cos‘𝐴))

Theoremsinmpi 23930 Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴))

Theoremcosmpi 23931 Cosine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 − π)) = -(cos‘𝐴))

Theoremsinppi 23932 Sine of a number plus π. (Contributed by NM, 10-Aug-2008.)
(𝐴 ∈ ℂ → (sin‘(𝐴 + π)) = -(sin‘𝐴))

Theoremcosppi 23933 Cosine of a complex number plus π. (Contributed by NM, 18-Aug-2008.)
(𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴))

Theoremefimpi 23934 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 23935 The sine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))

Theoremsinhalfpim 23936 The sine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))

Theoremcoshalfpip 23937 The cosine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) + 𝐴)) = -(sin‘𝐴))

Theoremcoshalfpim 23938 The cosine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
(𝐴 ∈ ℂ → (cos‘((π / 2) − 𝐴)) = (sin‘𝐴))

Theoremptolemy 23939 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 14610, 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 23940 Lemma for sincosq1sgn 23941. (Contributed by Paul Chapman, 24-Jan-2008.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴𝐴 < (π / 2)) → 0 < (sin‘𝐴))

Theoremsincosq1sgn 23941 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 23942 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 23943 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 23944 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 23945 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 23946 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 23947 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)

Theoremtangtx 23948 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 23949 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 23950 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 23951 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 23952 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 23953 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 23954 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 23955 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 23956 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 23957 The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.)
(tan‘(π / 4)) = 1

Theoremsincos6thpi 23958 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 23959 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 23960 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 23961 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 23962 The sine of an integer multiple of π is 0. (Contributed by NM, 11-Aug-2008.)
(𝐾 ∈ ℤ → (sin‘(𝐾 · π)) = 0)

Theoremcoskpi 23963 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 23964 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 23965 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 23966 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 23967 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 23968 Lemma for cosord 23969. (Contributed by Mario Carneiro, 10-May-2014.)
(𝜑𝐴 ∈ (0[,]π))    &   (𝜑𝐵 ∈ (0[,]π))    &   (𝜑𝐴 < 𝐵)       (𝜑 → (cos‘𝐵) < (cos‘𝐴))

Theoremcosord 23969 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 23970 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 23971 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 23972 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 23973 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 23974 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 23975.) (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 23975 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 23976 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 23977 (-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 23978* 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 23979* Lemma for efif1o 23983. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ (𝑥𝐷𝑦𝐷)) → (abs‘(𝑥𝑦)) < (2 · π))

Theoremefif1olem2 23980* Lemma for efif1o 23983. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ∃𝑦𝐷 ((𝑧𝑦) / (2 · π)) ∈ ℤ)

Theoremefif1olem3 23981* Lemma for efif1o 23983. (Contributed by Mario Carneiro, 8-May-2015.)
𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))    &   𝐶 = (abs “ {1})       ((𝜑𝑥𝐶) → (ℑ‘(√‘𝑥)) ∈ (-1[,]1))

Theoremefif1olem4 23982* 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 23983* 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 23984* 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 23985* 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 23986 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 23987* 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 23988* 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 23989 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 23990 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 23991 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 23992 Extend class notation with the natural logarithm function on complex numbers.
class log

Syntaxccxp 23993 Extend class notation with the complex power function.
class 𝑐

Definitiondf-log 23994 Define the natural logarithm function on complex numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function"). To obtain a function, only the principle value of the multivalued inverses of the exponential function, i.e. the inverse whose imaginary part lies in the interval (-pi, pi], see https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Paul Chapman, 21-Apr-2008.)
log = (exp ↾ (ℑ “ (-π(,]π)))

Definitiondf-cxp 23995* 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 23996 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 23997 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))

Theoremdflog2 23998 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)

Theoremrelogrn 23999 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)

Theoremlogrncn 24000 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)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
 Copyright terms: Public domain < Previous  Next >