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Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcxpexpz 24101 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 24102 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0) → (𝐴𝑐𝐵) = (𝐴𝐵))

Theoremlogcxp 24103 Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))

Theoremcxp0 24104 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (𝐴𝑐0) = 1)

Theoremcxp1 24105 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (𝐴𝑐1) = 𝐴)

Theorem1cxp 24106 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (1↑𝑐𝐴) = 1)

Theoremecxp 24107 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 24108 Closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) ∈ ℂ)

Theoremrecxpcl 24109 Real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐵 ∈ ℝ) → (𝐴𝑐𝐵) ∈ ℝ)

Theoremrpcxpcl 24110 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (𝐴𝑐𝐵) ∈ ℝ+)

Theoremcxpne0 24111 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 24112 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 24113 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 24114 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 24115 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 24116 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 24117 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐵 ∈ ℝ) → 0 ≤ (𝐴𝑐𝐵))

Theoremmulcxplem 24118 Lemma for mulcxp 24119. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (0↑𝑐𝐶) = ((𝐴𝑐𝐶) · (0↑𝑐𝐶)))

Theoremmulcxp 24119 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 24120 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴𝑐𝐵)))

Theoremdivcxp 24121 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 24122 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 24123 Product of exponents law for complex exponentiation. Variation on cxpmul 24122 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 24124 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 24125 Generalize cxpmul2 24123 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℤ)) → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝐶))

Theoremabscxp 24126 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 24127 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 24128 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 24129 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵𝐶 ↔ (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶)))

Theoremcxplea 24130 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.)
(((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵𝐶) → (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶))

Theoremcxple2 24131 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 24132 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 24133 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 24134 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 24135 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
(((𝐴 ∈ ℝ+𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵𝐶 ↔ (𝐴𝑐𝐶) ≤ (𝐴𝑐𝐵)))

Theoremcxpsqrtlem 24136 Lemma for cxpsqrt 24137. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)

Theoremcxpsqrt 24137 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 24138 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
(𝐴 ∈ ℝ+ → (log‘(√‘𝐴)) = ((log‘𝐴) / 2))

Theoremcxp0d 24139 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 24140 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴𝑐1) = 𝐴)

Theorem1cxpd 24141 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (1↑𝑐𝐴) = 1)

Theoremcxpcld 24142 Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝑐𝐵) ∈ ℂ)

Theoremcxpmul2d 24143 Product of exponents law for complex exponentiation. Variation on cxpmul 24122 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 24144 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 24145 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 24146 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 24147 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 24148 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 24149 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 24150 Generalize cxpmul2 24123 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 24151 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 24152 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 24153 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 24154 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 24155 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 24156 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 24157 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 24158 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 24159 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 24160 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 24161 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 24162 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 24163 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴𝑐𝐵)))

Theoremrpcxpcld 24164 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝑐𝐵) ∈ ℝ+)

Theoremlogcxpd 24165 Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))

Theoremcxplt3d 24166 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 24167 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 24168 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 24169* 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 24170* 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 24171 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 24172* 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 24173* Derivative of square root function. (Contributed by Brendan Leahy, 18-Dec-2018.)
𝐷 = (ℂ ∖ (-∞(,]0))       (ℂ D (𝑥𝐷 ↦ (√‘𝑥))) = (𝑥𝐷 ↦ (1 / (2 · (√‘𝑥))))

Theoremcxpcn 24174* 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 24175* 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 24176* Lemma for cxpcn3 24177. (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 24177* 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 24178 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
(√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)

Theoremsqrtcn 24179 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 24181 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 24182 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 24183 Property of an 𝑁-th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝑁 ∈ ℕ → ((-1↑𝑐(2 / 𝑁))↑𝑁) = 1)

Theoremroot1eq1 24184 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 24185 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 24186* 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 24187 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 24188 Symmetry of the natural logarithm range by negation. Lemma for logrec 24189. (Contributed by Saveliy Skresanov, 27-Dec-2016.)
((𝐴 ∈ ran log ∧ ¬ (ℑ‘𝐴) = π) → -𝐴 ∈ ran log)

Theoremlogrec 24189 Logarithm of a reciprocal changes sign. (Contributed by Saveliy Skresanov, 28-Dec-2016.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (ℑ‘(log‘𝐴)) ≠ π) → (log‘𝐴) = -(log‘(1 / 𝐴)))

14.3.5  Logarithms to an arbitrary base

Define "log using an arbitrary base" function and then prove some of its properties. Note that logb is generalized to an arbitrary base and arbitrary parameter in , but it doesn't accept infinities as arguments, unlike log.

Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log".

There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions (operations): (𝐵 logb 𝑋) where 𝐵 is the base and 𝑋 is the argument of the logarithm function. An alternative would be to support the notational form (( logb𝐵)‘𝑋); that looks a little more like traditional notation. Such a function ( logb𝐵) for a fixed base can be obtained in Metamath (without the need for a new definition) by the curry function: (curry logb𝐵), see logbmpt 24214, logbf 24215 and logbfval 24216.

Syntaxclogb 24190 Extend class notation to include the logarithm generalized to an arbitrary base.
class logb

Definitiondf-logb 24191* Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as (𝐵 logb 𝑋) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition is according to Wikipedia "Complex logarithm": https://en.wikipedia.org/wiki/Complex_logarithm#Logarithms_to_other_bases (10-Jun-2020). (Contributed by David A. Wheeler, 21-Jan-2017.)
logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))

Theoremlogbval 24192 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))

Theoremlogbcl 24193 General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) ∈ ℂ)

Theoremlogbid1 24194 General logarithm is 1 when base and arg match. Property 1(a) of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by David A. Wheeler, 22-Jul-2017.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 logb 𝐴) = 1)

Theoremlogb1 24195 The logarithm of 1 to an arbitrary base 𝐵 is 0. Property 1(b) of [Cohen4] p. 361. See log1 24024. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (𝐵 logb 1) = 0)

Theoremelogb 24196 The general logarithm of a number to the base being Euler's constant is the natural logarithm of the number. Put another way, using e as the base in logb is the same as log. Definition in [Cohen4] p. 352. (Contributed by David A. Wheeler, 17-Oct-2017.) (Revised by David A. Wheeler and AV, 16-Jun-2020.)
(𝐴 ∈ (ℂ ∖ {0}) → (e logb 𝐴) = (log‘𝐴))

Theoremlogbchbase 24197 Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐴 logb 𝑋) = ((𝐵 logb 𝑋) / (𝐵 logb 𝐴)))

Theoremrelogbval 24198 Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))

Theoremrelogbcl 24199 Closure of the general logarithm with a positive real base on positive reals. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
((𝐵 ∈ ℝ+𝑋 ∈ ℝ+𝐵 ≠ 1) → (𝐵 logb 𝑋) ∈ ℝ)

Theoremrelogbzcl 24200 Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.) (Proof shortened by AV, 9-Jun-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) ∈ ℝ)

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