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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtanrpcl 24301 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
(𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)

Theoremtangtx 24302 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 24303 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 24304 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 24305 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 24306 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 24307 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 24308 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 24309 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 24310 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 24311 The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.)
(tan‘(π / 4)) = 1

Theoremsincos6thpi 24312 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 24313 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 24314 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 24315 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 24316 The sine of an integer multiple of π is 0. (Contributed by NM, 11-Aug-2008.)
(𝐾 ∈ ℤ → (sin‘(𝐾 · π)) = 0)

Theoremcoskpi 24317 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 24318 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 24319 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 24320 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 24321 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 24322 Lemma for cosord 24323. (Contributed by Mario Carneiro, 10-May-2014.)
(𝜑𝐴 ∈ (0[,]π))    &   (𝜑𝐵 ∈ (0[,]π))    &   (𝜑𝐴 < 𝐵)       (𝜑 → (cos‘𝐵) < (cos‘𝐴))

Theoremcosord 24323 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 24324 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 24325 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 24326 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 24327 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 24328 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 24329.) (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 24329 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 24330 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 24331 (-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 24332* 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 24333* Lemma for efif1o 24337. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ (𝑥𝐷𝑦𝐷)) → (abs‘(𝑥𝑦)) < (2 · π))

Theoremefif1olem2 24334* Lemma for efif1o 24337. (Contributed by Mario Carneiro, 13-May-2014.)
𝐷 = (𝐴(,](𝐴 + (2 · π)))       ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ∃𝑦𝐷 ((𝑧𝑦) / (2 · π)) ∈ ℤ)

Theoremefif1olem3 24335* Lemma for efif1o 24337. (Contributed by Mario Carneiro, 8-May-2015.)
𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))    &   𝐶 = (abs “ {1})       ((𝜑𝑥𝐶) → (ℑ‘(√‘𝑥)) ∈ (-1[,]1))

Theoremefif1olem4 24336* 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 24337* 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 24338* 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 24339* 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 24340 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 24341* 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 24342* 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 24343 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 24344 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 24345 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 24346 Extend class notation with the natural logarithm function on complex numbers.
class log

Syntaxccxp 24347 Extend class notation with the complex power function.
class 𝑐

Definitiondf-log 24348 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 24349* 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 24350 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 24351 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 24352 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 24353 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 24354 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)

Theoremeff1o2 24355 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})

Theoremlogf1o 24356 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

Theoremdfrelog 24357 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)

Theoremrelogf1o 24358 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

Theoremlogrncl 24359 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ran log)

Theoremlogcl 24360 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)

Theoremlogimcl 24361 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))

Theoremlogcld 24362 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 24360. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (log‘𝑋) ∈ ℂ)

Theoremlogimcld 24363 The imaginary part of the logarithm is in (-π(,]π). Deduction form of logimcl 24361. Compare logimclad 24364. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (-π < (ℑ‘(log‘𝑋)) ∧ (ℑ‘(log‘𝑋)) ≤ π))

Theoremlogimclad 24364 The imaginary part of the logarithm is in (-π(,]π). Alternate form of logimcld 24363. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (ℑ‘(log‘𝑋)) ∈ (-π(,]π))

Theoremabslogimle 24365 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)

Theoremlogrnaddcl 24366 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)

Theoremrelogcl 24367 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)

Theoremeflog 24368 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)

Theoremlogeq0im1 24369 If the logarithm of a number is 0, the number must be 1. (Contributed by David A. Wheeler, 22-Jul-2017.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (log‘𝐴) = 0) → 𝐴 = 1)

Theoremlogccne0 24370 The logarithm isn't 0 if its argument isn't 0 or 1. (Contributed by David A. Wheeler, 17-Jul-2017.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ≠ 0)

Theoremlogne0 24371 Logarithm of a non-1 positive real number is not zero and thus suitable as a divisor. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 14-Jun-2020.)
((𝐴 ∈ ℝ+𝐴 ≠ 1) → (log‘𝐴) ≠ 0)

Theoremreeflog 24372 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴)

Theoremlogef 24373 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
(𝐴 ∈ ran log → (log‘(exp‘𝐴)) = 𝐴)

Theoremrelogef 24374 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(𝐴 ∈ ℝ → (log‘(exp‘𝐴)) = 𝐴)

Theoremlogeftb 24375 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ran log) → ((log‘𝐴) = 𝐵 ↔ (exp‘𝐵) = 𝐴))

Theoremrelogeftb 24376 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → ((log‘𝐴) = 𝐵 ↔ (exp‘𝐵) = 𝐴))

Theoremlog1 24377 The natural logarithm of 1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(log‘1) = 0

Theoremloge 24378 The natural logarithm of e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(log‘e) = 1

Theoremlogneg 24379 The natural logarithm of a negative real number. (Contributed by Mario Carneiro, 13-May-2014.) (Revised by Mario Carneiro, 3-Apr-2015.)
(𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π)))

Theoremlogm1 24380 The natural logarithm of negative 1. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
(log‘-1) = (i · π)

Theoremlognegb 24381 If a number has imaginary part equal to π, then it is on the negative real axis and vice-versa. (Contributed by Mario Carneiro, 23-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ+ ↔ (ℑ‘(log‘𝐴)) = π))

Theoremrelogoprlem 24382 Lemma for relogmul 24383 and relogdiv 24384. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2"). (Contributed by Steve Rodriguez, 25-Nov-2007.)
(((log‘𝐴) ∈ ℂ ∧ (log‘𝐵) ∈ ℂ) → (exp‘((log‘𝐴)𝐹(log‘𝐵))) = ((exp‘(log‘𝐴))𝐺(exp‘(log‘𝐵))))    &   (((log‘𝐴) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ) → ((log‘𝐴)𝐹(log‘𝐵)) ∈ ℝ)       ((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (log‘(𝐴𝐺𝐵)) = ((log‘𝐴)𝐹(log‘𝐵)))

Theoremrelogmul 24383 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵)))

Theoremrelogdiv 24384 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵)))

Theoremexplog 24385 Exponentiation of a nonzero complex number to an integer power. (Contributed by Paul Chapman, 21-Apr-2008.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) = (exp‘(𝑁 · (log‘𝐴))))

Theoremreexplog 24386 Exponentiation of a positive real number to an integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝑁 ∈ ℤ) → (𝐴𝑁) = (exp‘(𝑁 · (log‘𝐴))))

Theoremrelogexp 24387 The natural logarithm of positive 𝐴 raised to an integer power. Property 4 of [Cohen] p. 301-302, restricted to natural logarithms and integer powers 𝑁. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝑁 ∈ ℤ) → (log‘(𝐴𝑁)) = (𝑁 · (log‘𝐴)))

Theoremrelog 24388 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))

Theoremrelogiso 24389 The natural logarithm function on positive reals determines an isomorphism from the positive reals onto the reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(log ↾ ℝ+) Isom < , < (ℝ+, ℝ)

Theoremreloggim 24390 The natural logarithm is a group isomorphism from the group of positive reals under multiplication to the group of reals under addition. (Contributed by Mario Carneiro, 21-Jun-2015.) (Revised by Thierry Arnoux, 30-Jun-2019.)
𝑃 = ((mulGrp‘ℂfld) ↾s+)       (log ↾ ℝ+) ∈ (𝑃 GrpIso ℝfld)

Theoremlogltb 24391 The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (log‘𝐴) < (log‘𝐵)))

Theoremlogfac 24392* The logarithm of a factorial can be expressed as a finite sum of logs. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝑁 ∈ ℕ0 → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘))

Theoremeflogeq 24393* Solve an equation involving an exponential. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((exp‘𝐴) = 𝐵 ↔ ∃𝑛 ∈ ℤ 𝐴 = ((log‘𝐵) + ((i · (2 · π)) · 𝑛))))

Theoremlogleb 24394 Natural logarithm preserves . (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))

Theoremrplogcl 24395 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)
((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (log‘𝐴) ∈ ℝ+)

Theoremlogge0 24396 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (log‘𝐴))

Theoremlogcj 24397 The natural logarithm distributes under conjugation away from the branch cut. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ≠ 0) → (log‘(∗‘𝐴)) = (∗‘(log‘𝐴)))

Theoremefiarg 24398 The exponential of the "arg" function ℑ ∘ log. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))

Theoremcosargd 24399 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 24398. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = ((ℜ‘𝑋) / (abs‘𝑋)))

Theoremcosarg0d 24400 The cosine of the argument is zero precisely on the imaginary axis. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → ((cos‘(ℑ‘(log‘𝑋))) = 0 ↔ (ℜ‘𝑋) = 0))

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