P. 266 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26501-26600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cosord 26501	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 26502	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 26503	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 26504	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 26505	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 26506	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 26507	The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 26508.) (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 26508	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 26509	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 26510	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 26511*	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 26512*	Lemma for efif1o 26516. (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 26513*	Lemma for efif1o 26516. (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 26514*	Lemma for efif1o 26516. (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 26515*	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 26516*	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 26517*	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 26518*	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 26519	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 26520*	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 26521*	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 26522	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 26523	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 26524	Extend class notation with the natural logarithm function on complex numbers.
class log
Syntax	ccxp 26525	Extend class notation with the complex power function.
class ↑𝑐
Definition	df-log 26526	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 26527*	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 26528	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 26529	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 26530	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 26531	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 26532	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 26533	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 26534	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 26535	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 26536	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 26537	Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ran log)
Theorem	logcl 26538	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 26539	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 26540	The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 26538. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (log‘𝑋) ∈ ℂ)
Theorem	logimcld 26541	The imaginary part of the logarithm is in (-π(,]π). Deduction form of logimcl 26539. Compare logimclad 26542. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (-π < (ℑ‘(log‘𝑋)) ∧ (ℑ‘(log‘𝑋)) ≤ π))
Theorem	logimclad 26542	The imaginary part of the logarithm is in (-π(,]π). Alternate form of logimcld 26541. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (ℑ‘(log‘𝑋)) ∈ (-π(,]π))
Theorem	abslogimle 26543	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 26544	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 26545	Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)
Theorem	eflog 26546	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)
Theorem	logeq0im1 26547	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)
Theorem	logccne0 26548	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)
Theorem	logne0 26549	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)
Theorem	reeflog 26550	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ (𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴)
Theorem	logef 26551	Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
⊢ (𝐴 ∈ ran log → (log‘(exp‘𝐴)) = 𝐴)
Theorem	relogef 26552	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ (𝐴 ∈ ℝ → (log‘(exp‘𝐴)) = 𝐴)
Theorem	logeftb 26553	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))
Theorem	relogeftb 26554	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))
Theorem	log1 26555	The natural logarithm of 1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ (log‘1) = 0
Theorem	loge 26556	The natural logarithm of e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ (log‘e) = 1
Theorem	logi 26557	The natural logarithm of i. (Contributed by Scott Fenton, 13-Apr-2020.)
⊢ (log‘i) = (i · (π / 2))
Theorem	logneg 26558	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)))
Theorem	logm1 26559	The natural logarithm of negative 1. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
⊢ (log‘-1) = (i · π)
Theorem	lognegb 26560	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))
Theorem	relogoprlem 26561	Lemma for relogmul 26562 and relogdiv 26563. 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)))
Theorem	relogmul 26562	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)))
Theorem	relogdiv 26563	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)))
Theorem	explog 26564	Exponentiation of a nonzero complex number to an integer power. (Contributed by Paul Chapman, 21-Apr-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
Theorem	reexplog 26565	Exponentiation of a positive real number to an integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
Theorem	relogexp 26566	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‘(𝐴↑𝑁)) = (𝑁 · (log‘𝐴)))
Theorem	relog 26567	Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
Theorem	relogiso 26568	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)
Theorem	reloggim 26569	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 ℝ+)    ⇒   ⊢ (log ↾ ℝ+) ∈ (𝑃 GrpIso ℝfld)
Theorem	logltb 26570	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)))
Theorem	logfac 26571*	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))
Theorem	eflogeq 26572*	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))))
Theorem	logleb 26573	Natural logarithm preserves ≤. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 ≤ 𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))
Theorem	rplogcl 26574	Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (log‘𝐴) ∈ ℝ+)
Theorem	logge0 26575	The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (log‘𝐴))
Theorem	logcj 26576	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)))
Theorem	efiarg 26577	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)))
Theorem	cosargd 26578	The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 26577. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = ((ℜ‘𝑋) / (abs‘𝑋)))
Theorem	cosarg0d 26579	The cosine of the argument is zero precisely on the imaginary axis. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → ((cos‘(ℑ‘(log‘𝑋))) = 0 ↔ (ℜ‘𝑋) = 0))
Theorem	argregt0 26580	Closure of the argument of a complex number with positive real part. (Contributed by Mario Carneiro, 25-Feb-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (-(π / 2)(,)(π / 2)))
Theorem	argrege0 26581	Closure of the argument of a complex number with nonnegative real part. (Contributed by Mario Carneiro, 2-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)))
Theorem	argimgt0 26582	Closure of the argument of a complex number with positive imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (0(,)π))
Theorem	argimlt0 26583	Closure of the argument of a complex number with negative imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ (-π(,)0))
Theorem	logimul 26584	Multiplying a number by i increases the logarithm of the number by iπ / 2. (Contributed by Mario Carneiro, 4-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘(i · 𝐴)) = ((log‘𝐴) + (i · (π / 2))))
Theorem	logneg2 26585	The logarithm of the negative of a number with positive imaginary part is i · π less than the original. (Compare logneg 26558.) (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘-𝐴) = ((log‘𝐴) − (i · π)))
Theorem	logmul2 26586	Generalization of relogmul 26562 to a complex left argument. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵)))
Theorem	logdiv2 26587	Generalization of relogdiv 26563 to a complex left argument. (Contributed by Mario Carneiro, 8-Jul-2017.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵)))
Theorem	abslogle 26588	Bound on the magnitude of the complex logarithm function. (Contributed by Mario Carneiro, 3-Jul-2017.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(log‘𝐴)) ≤ ((abs‘(log‘(abs‘𝐴))) + π))
Theorem	tanarg 26589	The basic relation between the "arg" function ℑ ∘ log and the arctangent. (Contributed by Mario Carneiro, 25-Feb-2015.)
⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (tan‘(ℑ‘(log‘𝐴))) = ((ℑ‘𝐴) / (ℜ‘𝐴)))
Theorem	logdivlti 26590	The log𝑥 / 𝑥 function is strictly decreasing on the reals greater than e. (Contributed by Mario Carneiro, 14-Mar-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐴 < 𝐵) → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))
Theorem	logdivlt 26591	The log𝑥 / 𝑥 function is strictly decreasing on the reals greater than e. (Contributed by Mario Carneiro, 14-Mar-2014.)
⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
Theorem	logdivle 26592	The log𝑥 / 𝑥 function is strictly decreasing on the reals greater than e. (Contributed by Mario Carneiro, 3-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 ≤ 𝐵 ↔ ((log‘𝐵) / 𝐵) ≤ ((log‘𝐴) / 𝐴)))
Theorem	relogcld 26593	Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
Theorem	reeflogd 26594	Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (exp‘(log‘𝐴)) = 𝐴)
Theorem	relogmuld 26595	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 Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵)))
Theorem	relogdivd 26596	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 Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵)))
Theorem	logled 26597	Natural logarithm preserves ≤. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))
Theorem	relogefd 26598	Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (log‘(exp‘𝐴)) = 𝐴)
Theorem	rplogcld 26599	Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 1 < 𝐴)    ⇒   ⊢ (𝜑 → (log‘𝐴) ∈ ℝ+)
Theorem	logge0d 26600	The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 1 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 0 ≤ (log‘𝐴))