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Theorem List for Metamath Proof Explorer - 25101-25200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrelogoprlem 25101 Lemma for relogmul 25102 and relogdiv 25103. 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 25102 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 25103 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 25104 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 25105 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 25106 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 25107 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
 
Theoremrelogiso 25108 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 25109 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 25110 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 25111* 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 25112* 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 25113 Natural logarithm preserves . (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))
 
Theoremrplogcl 25114 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)
((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (log‘𝐴) ∈ ℝ+)
 
Theoremlogge0 25115 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 25116 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 25117 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 25118 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 25117. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = ((ℜ‘𝑋) / (abs‘𝑋)))
 
Theoremcosarg0d 25119 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))
 
Theoremargregt0 25120 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)))
 
Theoremargrege0 25121 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)))
 
Theoremargimgt0 25122 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))
 
Theoremargimlt0 25123 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))
 
Theoremlogimul 25124 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))))
 
Theoremlogneg2 25125 The logarithm of the negative of a number with positive imaginary part is i · π less than the original. (Compare logneg 25098.) (Contributed by Mario Carneiro, 3-Apr-2015.)
((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘-𝐴) = ((log‘𝐴) − (i · π)))
 
Theoremlogmul2 25126 Generalization of relogmul 25102 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)))
 
Theoremlogdiv2 25127 Generalization of relogdiv 25103 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)))
 
Theoremabslogle 25128 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))
 
Theoremtanarg 25129 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)))
 
Theoremlogdivlti 25130 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))
 
Theoremlogdivlt 25131 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)))
 
Theoremlogdivle 25132 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)) → (𝐴𝐵 ↔ ((log‘𝐵) / 𝐵) ≤ ((log‘𝐴) / 𝐴)))
 
Theoremrelogcld 25133 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (log‘𝐴) ∈ ℝ)
 
Theoremreeflogd 25134 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (exp‘(log‘𝐴)) = 𝐴)
 
Theoremrelogmuld 25135 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.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵)))
 
Theoremrelogdivd 25136 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.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵)))
 
Theoremlogled 25137 Natural logarithm preserves . (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))
 
Theoremrelogefd 25138 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (log‘(exp‘𝐴)) = 𝐴)
 
Theoremrplogcld 25139 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)       (𝜑 → (log‘𝐴) ∈ ℝ+)
 
Theoremlogge0d 25140 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → 0 ≤ (log‘𝐴))
 
Theoremlogge0b 25141 The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.)
(𝐴 ∈ ℝ+ → (0 ≤ (log‘𝐴) ↔ 1 ≤ 𝐴))
 
Theoremloggt0b 25142 The logarithm of a number is positive iff the number is greater than 1. (Contributed by AV, 30-May-2020.)
(𝐴 ∈ ℝ+ → (0 < (log‘𝐴) ↔ 1 < 𝐴))
 
Theoremlogle1b 25143 The logarithm of a number is less than or equal to 1 iff the number is less than or equal to Euler's constant. (Contributed by AV, 30-May-2020.)
(𝐴 ∈ ℝ+ → ((log‘𝐴) ≤ 1 ↔ 𝐴 ≤ e))
 
Theoremloglt1b 25144 The logarithm of a number is less than 1 iff the number is less than Euler's constant. (Contributed by AV, 30-May-2020.)
(𝐴 ∈ ℝ+ → ((log‘𝐴) < 1 ↔ 𝐴 < e))
 
Theoremdivlogrlim 25145 The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)
(𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
 
Theoremlogno1 25146 The logarithm function is not eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 30-May-2016.)
¬ (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) ∈ 𝑂(1)
 
Theoremdvrelog 25147 The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)
(ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))
 
Theoremrelogcn 25148 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)
(log ↾ ℝ+) ∈ (ℝ+cn→ℝ)
 
Theoremellogdm 25149 Elementhood in the "continuous domain" of the complex logarithm. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝐴𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)))
 
Theoremlogdmn0 25150 A number in the continuous domain of log is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝐴𝐷𝐴 ≠ 0)
 
Theoremlogdmnrp 25151 A number in the continuous domain of log is not a strictly negative number. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝐴𝐷 → ¬ -𝐴 ∈ ℝ+)
 
Theoremlogdmss 25152 The continuity domain of log is a subset of the regular domain of log. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       𝐷 ⊆ (ℂ ∖ {0})
 
Theoremlogcnlem2 25153 Lemma for logcn 25157. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝑆 = if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴)))    &   𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅)))    &   (𝜑𝐴𝐷)    &   (𝜑𝑅 ∈ ℝ+)       (𝜑 → if(𝑆𝑇, 𝑆, 𝑇) ∈ ℝ+)
 
Theoremlogcnlem3 25154 Lemma for logcn 25157. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝑆 = if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴)))    &   𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅)))    &   (𝜑𝐴𝐷)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐷)    &   (𝜑 → (abs‘(𝐴𝐵)) < if(𝑆𝑇, 𝑆, 𝑇))       (𝜑 → (-π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∧ ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π))
 
Theoremlogcnlem4 25155 Lemma for logcn 25157. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝑆 = if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴)))    &   𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅)))    &   (𝜑𝐴𝐷)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐷)    &   (𝜑 → (abs‘(𝐴𝐵)) < if(𝑆𝑇, 𝑆, 𝑇))       (𝜑 → (abs‘((ℑ‘(log‘𝐴)) − (ℑ‘(log‘𝐵)))) < 𝑅)
 
Theoremlogcnlem5 25156* Lemma for logcn 25157. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝑥𝐷 ↦ (ℑ‘(log‘𝑥))) ∈ (𝐷cn→ℝ)
 
Theoremlogcn 25157 The logarithm function is continuous away from the branch cut at negative reals. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (log ↾ 𝐷) ∈ (𝐷cn→ℂ)
 
Theoremdvloglem 25158 Lemma for dvlog 25161. (Contributed by Mario Carneiro, 24-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (log “ 𝐷) ∈ (TopOpen‘ℂfld)
 
Theoremlogdmopn 25159 The "continuous domain" of log is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       𝐷 ∈ (TopOpen‘ℂfld)
 
Theoremlogf1o2 25160 The logarithm maps its continuous domain bijectively onto the set of numbers with imaginary part -π < ℑ(𝑧) < π. The negative reals are mapped to the numbers with imaginary part equal to π. (Contributed by Mario Carneiro, 2-May-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (log ↾ 𝐷):𝐷1-1-onto→(ℑ “ (-π(,)π))
 
Theoremdvlog 25161* The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (ℂ D (log ↾ 𝐷)) = (𝑥𝐷 ↦ (1 / 𝑥))
 
Theoremdvlog2lem 25162 Lemma for dvlog2 25163. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       𝑆 ⊆ (ℂ ∖ (-∞(,]0))
 
Theoremdvlog2 25163* The derivative of the complex logarithm function on the open unit ball centered at 1, a sometimes easier region to work with than the ℂ ∖ (-∞, 0] of dvlog 25161. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       (ℂ D (log ↾ 𝑆)) = (𝑥𝑆 ↦ (1 / 𝑥))
 
Theoremadvlog 25164 The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.)
(ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))
 
Theoremadvlogexp 25165* The antiderivative of a power of the logarithm. (Set 𝐴 = 1 and multiply by (-1)↑𝑁 · 𝑁! to get the antiderivative of log(𝑥)↑𝑁 itself.) (Contributed by Mario Carneiro, 22-May-2016.)
((𝐴 ∈ ℝ+𝑁 ∈ ℕ0) → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝐴 / 𝑥))↑𝑘) / (!‘𝑘))))) = (𝑥 ∈ ℝ+ ↦ (((log‘(𝐴 / 𝑥))↑𝑁) / (!‘𝑁))))
 
Theoremefopnlem1 25166 Lemma for efopn 25168. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
(((𝑅 ∈ ℝ+𝑅 < π) ∧ 𝐴 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (abs‘(ℑ‘𝐴)) < π)
 
Theoremefopnlem2 25167 Lemma for efopn 25168. (Contributed by Mario Carneiro, 2-May-2015.)
𝐽 = (TopOpen‘ℂfld)       ((𝑅 ∈ ℝ+𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)
 
Theoremefopn 25168 The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐽 = (TopOpen‘ℂfld)       (𝑆𝐽 → (exp “ 𝑆) ∈ 𝐽)
 
Theoremlogtayllem 25169* Lemma for logtayl 25170. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))) ∈ dom ⇝ )
 
Theoremlogtayl 25170* The Taylor series for -log(1 − 𝐴). (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))) ⇝ -(log‘(1 − 𝐴)))
 
Theoremlogtaylsum 25171* The Taylor series for -log(1 − 𝐴), as an infinite sum. (Contributed by Mario Carneiro, 31-Mar-2015.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → Σ𝑘 ∈ ℕ ((𝐴𝑘) / 𝑘) = -(log‘(1 − 𝐴)))
 
Theoremlogtayl2 25172* Power series expression for the logarithm. (Contributed by Mario Carneiro, 31-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       (𝐴𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))) ⇝ (log‘𝐴))
 
Theoremlogccv 25173 The natural logarithm function on the reals is a strictly concave function. (Contributed by Mario Carneiro, 20-Jun-2015.)
(((𝐴 ∈ ℝ+𝐵 ∈ ℝ+𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) < (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
 
Theoremcxpval 25174 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))))
 
Theoremcxpef 25175 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
 
Theorem0cxp 25176 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0)
 
Theoremcxpexpz 25177 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ) → (𝐴𝑐𝐵) = (𝐴𝐵))
 
Theoremcxpexp 25178 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0) → (𝐴𝑐𝐵) = (𝐴𝐵))
 
Theoremlogcxp 25179 Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))
 
Theoremcxp0 25180 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (𝐴𝑐0) = 1)
 
Theoremcxp1 25181 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (𝐴𝑐1) = 𝐴)
 
Theorem1cxp 25182 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (1↑𝑐𝐴) = 1)
 
Theoremecxp 25183 Write the exponential function as an exponent to the power e. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (e↑𝑐𝐴) = (exp‘𝐴))
 
Theoremcxpcl 25184 Closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) ∈ ℂ)
 
Theoremrecxpcl 25185 Real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐵 ∈ ℝ) → (𝐴𝑐𝐵) ∈ ℝ)
 
Theoremrpcxpcl 25186 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (𝐴𝑐𝐵) ∈ ℝ+)
 
Theoremcxpne0 25187 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) ≠ 0)
 
Theoremcxpeq0 25188 Complex exponentiation is zero iff the mantissa is zero and the exponent is nonzero. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝑐𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 ≠ 0)))
 
Theoremcxpadd 25189 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴𝑐(𝐵 + 𝐶)) = ((𝐴𝑐𝐵) · (𝐴𝑐𝐶)))
 
Theoremcxpp1 25190 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴𝑐(𝐵 + 1)) = ((𝐴𝑐𝐵) · 𝐴))
 
Theoremcxpneg 25191 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴𝑐-𝐵) = (1 / (𝐴𝑐𝐵)))
 
Theoremcxpsub 25192 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴𝑐(𝐵𝐶)) = ((𝐴𝑐𝐵) / (𝐴𝑐𝐶)))
 
Theoremcxpge0 25193 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐵 ∈ ℝ) → 0 ≤ (𝐴𝑐𝐵))
 
Theoremmulcxplem 25194 Lemma for mulcxp 25195. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (0↑𝑐𝐶) = ((𝐴𝑐𝐶) · (0↑𝑐𝐶)))
 
Theoremmulcxp 25195 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴𝑐𝐶) · (𝐵𝑐𝐶)))
 
Theoremcxprec 25196 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴𝑐𝐵)))
 
Theoremdivcxp 25197 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+𝐶 ∈ ℂ) → ((𝐴 / 𝐵)↑𝑐𝐶) = ((𝐴𝑐𝐶) / (𝐵𝑐𝐶)))
 
Theoremcxpmul 25198 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ) → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝑐𝐶))
 
Theoremcxpmul2 25199 Product of exponents law for complex exponentiation. Variation on cxpmul 25198 with more general conditions on 𝐴 and 𝐵 when 𝐶 is an integer. (Contributed by Mario Carneiro, 9-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0) → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝐶))
 
Theoremcxproot 25200 The complex power function allows us to write n-th roots via the idiom 𝐴𝑐(1 / 𝑁). (Contributed by Mario Carneiro, 6-May-2015.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑐(1 / 𝑁))↑𝑁) = 𝐴)
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