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Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremadvlogexp 24201* 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 24202 Lemma for efopn 24204. (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 24203 Lemma for efopn 24204. (Contributed by Mario Carneiro, 2-May-2015.)
𝐽 = (TopOpen‘ℂfld)       ((𝑅 ∈ ℝ+𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)

Theoremefopn 24204 The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐽 = (TopOpen‘ℂfld)       (𝑆𝐽 → (exp “ 𝑆) ∈ 𝐽)

Theoremlogtayllem 24205* Lemma for logtayl 24206. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))) ∈ dom ⇝ )

Theoremlogtayl 24206* 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 24207* 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 24208* 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 24209 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 24210 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 24211 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 24212 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 24213 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 24214 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0) → (𝐴𝑐𝐵) = (𝐴𝐵))

Theoremlogcxp 24215 Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))

Theoremcxp0 24216 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (𝐴𝑐0) = 1)

Theoremcxp1 24217 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (𝐴𝑐1) = 𝐴)

Theorem1cxp 24218 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (1↑𝑐𝐴) = 1)

Theoremecxp 24219 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 24220 Closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) ∈ ℂ)

Theoremrecxpcl 24221 Real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐵 ∈ ℝ) → (𝐴𝑐𝐵) ∈ ℝ)

Theoremrpcxpcl 24222 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (𝐴𝑐𝐵) ∈ ℝ+)

Theoremcxpne0 24223 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 24224 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 24225 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 24226 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 24227 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 24228 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 24229 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐵 ∈ ℝ) → 0 ≤ (𝐴𝑐𝐵))

Theoremmulcxplem 24230 Lemma for mulcxp 24231. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (0↑𝑐𝐶) = ((𝐴𝑐𝐶) · (0↑𝑐𝐶)))

Theoremmulcxp 24231 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 24232 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴𝑐𝐵)))

Theoremdivcxp 24233 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 24234 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 24235 Product of exponents law for complex exponentiation. Variation on cxpmul 24234 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 24236 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 24237 Generalize cxpmul2 24235 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℤ)) → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝐶))

Theoremabscxp 24238 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 24239 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 24240 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 24241 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵𝐶 ↔ (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶)))

Theoremcxplea 24242 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.)
(((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵𝐶) → (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶))

Theoremcxple2 24243 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 24244 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 24245 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 24246 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 24247 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
(((𝐴 ∈ ℝ+𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵𝐶 ↔ (𝐴𝑐𝐶) ≤ (𝐴𝑐𝐵)))

Theoremcxpsqrtlem 24248 Lemma for cxpsqrt 24249. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)

Theoremcxpsqrt 24249 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 24250 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
(𝐴 ∈ ℝ+ → (log‘(√‘𝐴)) = ((log‘𝐴) / 2))

Theoremcxp0d 24251 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 24252 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴𝑐1) = 𝐴)

Theorem1cxpd 24253 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (1↑𝑐𝐴) = 1)

Theoremcxpcld 24254 Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝑐𝐵) ∈ ℂ)

Theoremcxpmul2d 24255 Product of exponents law for complex exponentiation. Variation on cxpmul 24234 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))

Theorem0cxpd 24256 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (0↑𝑐𝐴) = 0)

Theoremcxpexpzd 24257 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℤ)       (𝜑 → (𝐴𝑐𝐵) = (𝐴𝐵))

Theoremcxpefd 24258 Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))

Theoremcxpne0d 24259 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝑐𝐵) ≠ 0)

Theoremcxpp1d 24260 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝑐(𝐵 + 1)) = ((𝐴𝑐𝐵) · 𝐴))

Theoremcxpnegd 24261 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝑐-𝐵) = (1 / (𝐴𝑐𝐵)))

Theoremcxpmul2zd 24262 Generalize cxpmul2 24235 to negative integers. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℤ)       (𝜑 → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝐶))

Theoremcxpaddd 24263 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴𝑐(𝐵 + 𝐶)) = ((𝐴𝑐𝐵) · (𝐴𝑐𝐶)))

Theoremcxpsubd 24264 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴𝑐(𝐵𝐶)) = ((𝐴𝑐𝐵) / (𝐴𝑐𝐶)))

Theoremcxpltd 24265 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵 < 𝐶 ↔ (𝐴𝑐𝐵) < (𝐴𝑐𝐶)))

Theoremcxpled 24266 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵𝐶 ↔ (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶)))

Theoremcxplead 24267 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶))

Theoremdivcxpd 24268 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 / 𝐵)↑𝑐𝐶) = ((𝐴𝑐𝐶) / (𝐵𝑐𝐶)))

Theoremrecxpcld 24269 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝑐𝐵) ∈ ℝ)

Theoremcxpge0d 24270 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴𝑐𝐵))

Theoremcxple2ad 24271 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝑐𝐶) ≤ (𝐵𝑐𝐶))

Theoremcxplt2d 24272 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴𝑐𝐶) < (𝐵𝑐𝐶)))

Theoremcxple2d 24273 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝑐𝐶) ≤ (𝐵𝑐𝐶)))

Theoremmulcxpd 24274 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴𝑐𝐶) · (𝐵𝑐𝐶)))

Theoremcxprecd 24275 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴𝑐𝐵)))

Theoremrpcxpcld 24276 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝑐𝐵) ∈ ℝ+)

Theoremlogcxpd 24277 Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))

Theoremcxplt3d 24278 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 1)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵 < 𝐶 ↔ (𝐴𝑐𝐶) < (𝐴𝑐𝐵)))

Theoremcxple3d 24279 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 1)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵𝐶 ↔ (𝐴𝑐𝐶) ≤ (𝐴𝑐𝐵)))

Theoremcxpmuld 24280 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝑐𝐶))

Theoremdvcxp1 24281* The derivative of a complex power with respect to the first argument. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥𝑐𝐴))) = (𝑥 ∈ ℝ+ ↦ (𝐴 · (𝑥𝑐(𝐴 − 1)))))

Theoremdvcxp2 24282* The derivative of a complex power with respect to the second argument. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝐴 ∈ ℝ+ → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴𝑐𝑥))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · (𝐴𝑐𝑥))))

Theoremdvsqrt 24283 The derivative of the real square root function. (Contributed by Mario Carneiro, 1-May-2016.)
(ℝ D (𝑥 ∈ ℝ+ ↦ (√‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / (2 · (√‘𝑥))))

Theoremdvcncxp1 24284* Derivative of complex power with respect to first argument on the complex plane. (Contributed by Brendan Leahy, 18-Dec-2018.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝐴 ∈ ℂ → (ℂ D (𝑥𝐷 ↦ (𝑥𝑐𝐴))) = (𝑥𝐷 ↦ (𝐴 · (𝑥𝑐(𝐴 − 1)))))

Theoremdvcnsqrt 24285* Derivative of square root function. (Contributed by Brendan Leahy, 18-Dec-2018.)
𝐷 = (ℂ ∖ (-∞(,]0))       (ℂ D (𝑥𝐷 ↦ (√‘𝑥))) = (𝑥𝐷 ↦ (1 / (2 · (√‘𝑥))))

Theoremcxpcn 24286* Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝐷)       (𝑥𝐷, 𝑦 ∈ ℂ ↦ (𝑥𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)

Theoremcxpcn2 24287* Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t+)       (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (𝑥𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)

Theoremcxpcn3lem 24288* Lemma for cxpcn3 24289. (Contributed by Mario Carneiro, 2-May-2016.)
𝐷 = (ℜ “ ℝ+)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t (0[,)+∞))    &   𝐿 = (𝐽t 𝐷)    &   𝑈 = (if((ℜ‘𝐴) ≤ 1, (ℜ‘𝐴), 1) / 2)    &   𝑇 = if(𝑈 ≤ (𝐸𝑐(1 / 𝑈)), 𝑈, (𝐸𝑐(1 / 𝑈)))       ((𝐴𝐷𝐸 ∈ ℝ+) → ∃𝑑 ∈ ℝ+𝑎 ∈ (0[,)+∞)∀𝑏𝐷 (((abs‘𝑎) < 𝑑 ∧ (abs‘(𝐴𝑏)) < 𝑑) → (abs‘(𝑎𝑐𝑏)) < 𝐸))

Theoremcxpcn3 24289* Extend continuity of the complex power function to a base of zero, as long as the exponent has strictly positive real part. (Contributed by Mario Carneiro, 2-May-2016.)
𝐷 = (ℜ “ ℝ+)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t (0[,)+∞))    &   𝐿 = (𝐽t 𝐷)       (𝑥 ∈ (0[,)+∞), 𝑦𝐷 ↦ (𝑥𝑐𝑦)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)

Theoremresqrtcn 24290 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
(√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)

Theoremsqrtcn 24291 Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)
𝐷 = (ℂ ∖ (-∞(,]0))       (√ ↾ 𝐷) ∈ (𝐷cn→ℂ)

(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 ≤ 1)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐵 ≤ 1)       (𝜑𝐴 ≤ (𝐴𝑐𝐵))

Theoremcxpaddle 24293 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐶 ≤ 1)       (𝜑 → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴𝑐𝐶) + (𝐵𝑐𝐶)))

Theoremabscxpbnd 24294 Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → 0 ≤ (ℜ‘𝐵))    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → (abs‘𝐴) ≤ 𝑀)       (𝜑 → (abs‘(𝐴𝑐𝐵)) ≤ ((𝑀𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))

Theoremroot1id 24295 Property of an 𝑁-th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝑁 ∈ ℕ → ((-1↑𝑐(2 / 𝑁))↑𝑁) = 1)

Theoremroot1eq1 24296 The only powers of an 𝑁-th root of unity that equal 1 are the multiples of 𝑁. In other words, -1↑𝑐(2 / 𝑁) has order 𝑁 in the multiplicative group of nonzero complex numbers. (In fact, these and their powers are the only elements of finite order in the complex numbers.) (Contributed by Mario Carneiro, 28-Apr-2016.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (((-1↑𝑐(2 / 𝑁))↑𝐾) = 1 ↔ 𝑁𝐾))

Theoremroot1cj 24297 Within the 𝑁-th roots of unity, the conjugate of the 𝐾-th root is the 𝑁𝐾-th root. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) = ((-1↑𝑐(2 / 𝑁))↑(𝑁𝐾)))

Theoremcxpeq 24298* Solve an equation involving an 𝑁-th power. The expression -1↑𝑐(2 / 𝑁) = exp(2πi / 𝑁) is a way to write the primitive 𝑁-th root of unity with the smallest positive argument. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → ((𝐴𝑁) = 𝐵 ↔ ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵𝑐(1 / 𝑁)) · ((-1↑𝑐(2 / 𝑁))↑𝑛))))

Theoremloglesqrt 24299 An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by AV, 2-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ≤ (√‘𝐴))

Theoremlogreclem 24300 Symmetry of the natural logarithm range by negation. Lemma for logrec 24301. (Contributed by Saveliy Skresanov, 27-Dec-2016.)
((𝐴 ∈ ran log ∧ ¬ (ℑ‘𝐴) = π) → -𝐴 ∈ ran log)

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