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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremargregt0 24401 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 24402 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 24403 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 24404 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 24405 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 24406 The logarithm of the negative of a number with positive imaginary part is i · π less than the original. (Compare logneg 24379.) (Contributed by Mario Carneiro, 3-Apr-2015.)
((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘-𝐴) = ((log‘𝐴) − (i · π)))

Theoremlogmul2 24407 Generalization of relogmul 24383 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 24408 Generalization of relogdiv 24384 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 24409 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 24410 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 24411 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 24412 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 24413 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 24414 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (log‘𝐴) ∈ ℝ)

Theoremreeflogd 24415 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 24416 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 24417 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 24418 Natural logarithm preserves . (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))

Theoremrelogefd 24419 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 24420 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 24421 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 24422 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 24423 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 24424 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 24425 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 24426 The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)
(𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0

Theoremlogno1 24427 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 24428 The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)
(ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))

Theoremrelogcn 24429 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)
(log ↾ ℝ+) ∈ (ℝ+cn→ℝ)

Theoremellogdm 24430 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 24431 A number in the continuous domain of log is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝐴𝐷𝐴 ≠ 0)

Theoremlogdmnrp 24432 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 24433 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 24434 Lemma for logcn 24438. (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 24435 Lemma for logcn 24438. (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 24436 Lemma for logcn 24438. (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 24437* Lemma for logcn 24438. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝑥𝐷 ↦ (ℑ‘(log‘𝑥))) ∈ (𝐷cn→ℝ)

Theoremlogcn 24438 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 24439 Lemma for dvlog 24442. (Contributed by Mario Carneiro, 24-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (log “ 𝐷) ∈ (TopOpen‘ℂfld)

Theoremlogdmopn 24440 The "continuous domain" of log is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       𝐷 ∈ (TopOpen‘ℂfld)

Theoremlogf1o2 24441 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 24442* 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 24443 Lemma for dvlog2 24444. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       𝑆 ⊆ (ℂ ∖ (-∞(,]0))

Theoremdvlog2 24444* 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 24442. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       (ℂ D (log ↾ 𝑆)) = (𝑥𝑆 ↦ (1 / 𝑥))

Theoremadvlog 24445 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 24446* 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 24447 Lemma for efopn 24449. (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 24448 Lemma for efopn 24449. (Contributed by Mario Carneiro, 2-May-2015.)
𝐽 = (TopOpen‘ℂfld)       ((𝑅 ∈ ℝ+𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)

Theoremefopn 24449 The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐽 = (TopOpen‘ℂfld)       (𝑆𝐽 → (exp “ 𝑆) ∈ 𝐽)

Theoremlogtayllem 24450* Lemma for logtayl 24451. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))) ∈ dom ⇝ )

Theoremlogtayl 24451* 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 24452* 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 24453* 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 24454 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 24455 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 24456 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 24457 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 24458 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 24459 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0) → (𝐴𝑐𝐵) = (𝐴𝐵))

Theoremlogcxp 24460 Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))

Theoremcxp0 24461 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (𝐴𝑐0) = 1)

Theoremcxp1 24462 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (𝐴𝑐1) = 𝐴)

Theorem1cxp 24463 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (1↑𝑐𝐴) = 1)

Theoremecxp 24464 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 24465 Closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) ∈ ℂ)

Theoremrecxpcl 24466 Real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐵 ∈ ℝ) → (𝐴𝑐𝐵) ∈ ℝ)

Theoremrpcxpcl 24467 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (𝐴𝑐𝐵) ∈ ℝ+)

Theoremcxpne0 24468 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 24469 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 24470 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 24471 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 24472 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 24473 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 24474 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐵 ∈ ℝ) → 0 ≤ (𝐴𝑐𝐵))

Theoremmulcxplem 24475 Lemma for mulcxp 24476. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (0↑𝑐𝐶) = ((𝐴𝑐𝐶) · (0↑𝑐𝐶)))

Theoremmulcxp 24476 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 24477 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴𝑐𝐵)))

Theoremdivcxp 24478 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 24479 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 24480 Product of exponents law for complex exponentiation. Variation on cxpmul 24479 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 24481 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)

Theoremcxpmul2z 24482 Generalize cxpmul2 24480 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℤ)) → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝐶))

Theoremabscxp 24483 Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → (abs‘(𝐴𝑐𝐵)) = (𝐴𝑐(ℜ‘𝐵)))

Theoremabscxp2 24484 Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴𝑐𝐵)) = ((abs‘𝐴)↑𝑐𝐵))

Theoremcxplt 24485 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴𝑐𝐵) < (𝐴𝑐𝐶)))

Theoremcxple 24486 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵𝐶 ↔ (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶)))

Theoremcxplea 24487 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.)
(((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵𝐶) → (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶))

Theoremcxple2 24488 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → (𝐴𝐵 ↔ (𝐴𝑐𝐶) ≤ (𝐵𝑐𝐶)))

Theoremcxplt2 24489 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴𝑐𝐶) < (𝐵𝑐𝐶)))

Theoremcxple2a 24490 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴𝐵) → (𝐴𝑐𝐶) ≤ (𝐵𝑐𝐶))

Theoremcxplt3 24491 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
(((𝐴 ∈ ℝ+𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴𝑐𝐶) < (𝐴𝑐𝐵)))

Theoremcxple3 24492 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
(((𝐴 ∈ ℝ+𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵𝐶 ↔ (𝐴𝑐𝐶) ≤ (𝐴𝑐𝐵)))

Theoremcxpsqrtlem 24493 Lemma for cxpsqrt 24494. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)

Theoremcxpsqrt 24494 The complex exponential function with exponent 1 / 2 exactly matches the complex square root function (the branch cut is in the same place for both functions), and thus serves as a suitable generalization to other 𝑛-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (𝐴𝑐(1 / 2)) = (√‘𝐴))

Theoremlogsqrt 24495 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
(𝐴 ∈ ℝ+ → (log‘(√‘𝐴)) = ((log‘𝐴) / 2))

Theoremcxp0d 24496 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴𝑐0) = 1)

Theoremcxp1d 24497 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴𝑐1) = 𝐴)

Theorem1cxpd 24498 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (1↑𝑐𝐴) = 1)

Theoremcxpcld 24499 Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝑐𝐵) ∈ ℂ)

Theoremcxpmul2d 24500 Product of exponents law for complex exponentiation. Variation on cxpmul 24479 with more general conditions on 𝐴 and 𝐵 when 𝐶 is an integer. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℕ0)       (𝜑 → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝐶))

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