P. 263 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cxprec 26201	Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴↑𝑐𝐵)))
Theorem	divcxp 26202	Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) / (𝐵↑𝑐𝐶)))
Theorem	cxpmul 26203	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) → (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝑐𝐶))
Theorem	cxpmul2 26204	Product of exponents law for complex exponentiation. Variation on cxpmul 26203 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) → (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝐶))
Theorem	cxproot 26205	The complex power function allows to write n-th roots via the idiom 𝐴↑𝑐(1 / 𝑁). (Contributed by Mario Carneiro, 6-May-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑐(1 / 𝑁))↑𝑁) = 𝐴)
Theorem	cxpmul2z 26206	Generalize cxpmul2 26204 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℤ)) → (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝐶))
Theorem	abscxp 26207	Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴↑𝑐𝐵)) = (𝐴↑𝑐(ℜ‘𝐵)))
Theorem	abscxp2 26208	Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴↑𝑐𝐵)) = ((abs‘𝐴)↑𝑐𝐵))
Theorem	cxplt 26209	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐵) < (𝐴↑𝑐𝐶)))
Theorem	cxple 26210	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 ≤ 𝐶 ↔ (𝐴↑𝑐𝐵) ≤ (𝐴↑𝑐𝐶)))
Theorem	cxplea 26211	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ≤ 𝐶) → (𝐴↑𝑐𝐵) ≤ (𝐴↑𝑐𝐶))
Theorem	cxple2 26212	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → (𝐴 ≤ 𝐵 ↔ (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶)))
Theorem	cxplt2 26213	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴↑𝑐𝐶) < (𝐵↑𝑐𝐶)))
Theorem	cxple2a 26214	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐶) ∧ 𝐴 ≤ 𝐵) → (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶))
Theorem	cxplt3 26215	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐶) < (𝐴↑𝑐𝐵)))
Theorem	cxple3 26216	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 ≤ 𝐶 ↔ (𝐴↑𝑐𝐶) ≤ (𝐴↑𝑐𝐵)))
Theorem	cxpsqrtlem 26217	Lemma for cxpsqrt 26218. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑐(1 / 2)) = -(√‘𝐴)) → (i · (√‘𝐴)) ∈ ℝ)
Theorem	cxpsqrt 26218	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 → (𝐴↑𝑐(1 / 2)) = (√‘𝐴))
Theorem	logsqrt 26219	Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
⊢ (𝐴 ∈ ℝ+ → (log‘(√‘𝐴)) = ((log‘𝐴) / 2))
Theorem	cxp0d 26220	Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐0) = 1)
Theorem	cxp1d 26221	Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐1) = 𝐴)
Theorem	1cxpd 26222	Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (1↑𝑐𝐴) = 1)
Theorem	cxpcld 26223	Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐𝐵) ∈ ℂ)
Theorem	cxpmul2d 26224	Product of exponents law for complex exponentiation. Variation on cxpmul 26203 with more general conditions on 𝐴 and 𝐵 when 𝐶 is an integer. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝐶))
Theorem	0cxpd 26225	Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (0↑𝑐𝐴) = 0)
Theorem	cxpexpzd 26226	Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐𝐵) = (𝐴↑𝐵))
Theorem	cxpefd 26227	Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
Theorem	cxpne0d 26228	Complex exponentiation is nonzero if its base is nonzero. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐𝐵) ≠ 0)
Theorem	cxpp1d 26229	Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐(𝐵 + 1)) = ((𝐴↑𝑐𝐵) · 𝐴))
Theorem	cxpnegd 26230	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐-𝐵) = (1 / (𝐴↑𝑐𝐵)))
Theorem	cxpmul2zd 26231	Generalize cxpmul2 26204 to negative integers. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝐶))
Theorem	cxpaddd 26232	Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐(𝐵 + 𝐶)) = ((𝐴↑𝑐𝐵) · (𝐴↑𝑐𝐶)))
Theorem	cxpsubd 26233	Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐(𝐵 − 𝐶)) = ((𝐴↑𝑐𝐵) / (𝐴↑𝑐𝐶)))
Theorem	cxpltd 26234	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 1 < 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐵) < (𝐴↑𝑐𝐶)))
Theorem	cxpled 26235	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 1 < 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐵 ≤ 𝐶 ↔ (𝐴↑𝑐𝐵) ≤ (𝐴↑𝑐𝐶)))
Theorem	cxplead 26236	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 1 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐𝐵) ≤ (𝐴↑𝑐𝐶))
Theorem	divcxpd 26237	Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 / 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) / (𝐵↑𝑐𝐶)))
Theorem	recxpcld 26238	Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐𝐵) ∈ ℝ)
Theorem	cxpge0d 26239	Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴↑𝑐𝐵))
Theorem	cxple2ad 26240	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶))
Theorem	cxplt2d 26241	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴↑𝑐𝐶) < (𝐵↑𝑐𝐶)))
Theorem	cxple2d 26242	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴↑𝑐𝐶) ≤ (𝐵↑𝑐𝐶)))
Theorem	mulcxpd 26243	Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶) · (𝐵↑𝑐𝐶)))
Theorem	recxpf1lem 26244	Complex exponentiation on positive real numbers is a one-to-one function. (Contributed by Thierry Arnoux, 1-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴↑𝑐𝐶) = (𝐵↑𝑐𝐶)))
Theorem	cxpsqrtth 26245	Square root theorem over the complex numbers for the complex power function. Theorem I.35 of [Apostol] p. 29. Compare with sqrtth 15313. (Contributed by AV, 23-Dec-2022.)
⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑𝑐2) = 𝐴)
Theorem	2irrexpq 26246*	There exist irrational numbers 𝑎 and 𝑏 such that (𝑎↑𝑐𝑏) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "classical proof" for theorem 1.2 of [Bauer], p. 483. This proof is not acceptable in intuitionistic logic, since it is based on the law of excluded middle: Either ((√‘2)↑𝑐(√‘2)) is rational, in which case (√‘2), being irrational (see sqrt2irr 16194), can be chosen for both 𝑎 and 𝑏, or ((√‘2)↑𝑐(√‘2)) is irrational, in which case ((√‘2)↑𝑐(√‘2)) can be chosen for 𝑎 and (√‘2) for 𝑏, since (((√‘2)↑𝑐(√‘2))↑𝑐(√‘2)) = 2 is rational. For an alternate proof, which can be used in intuitionistic logic, see 2irrexpqALT 26312. (Contributed by AV, 23-Dec-2022.)
⊢ ∃𝑎 ∈ (ℝ ∖ ℚ)∃𝑏 ∈ (ℝ ∖ ℚ)(𝑎↑𝑐𝑏) ∈ ℚ
Theorem	cxprecd 26247	Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴↑𝑐𝐵)))
Theorem	rpcxpcld 26248	Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐𝐵) ∈ ℝ+)
Theorem	logcxpd 26249	Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (log‘(𝐴↑𝑐𝐵)) = (𝐵 · (log‘𝐴)))
Theorem	cxplt3d 26250	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐵 < 𝐶 ↔ (𝐴↑𝑐𝐶) < (𝐴↑𝑐𝐵)))
Theorem	cxple3d 26251	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐵 ≤ 𝐶 ↔ (𝐴↑𝑐𝐶) ≤ (𝐴↑𝑐𝐵)))
Theorem	cxpmuld 26252	Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝑐𝐶))
Theorem	cxpgt0d 26253	A positive real raised to a real power is positive. (Contributed by SN, 6-Apr-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℝ)    ⇒   ⊢ (𝜑 → 0 < (𝐴↑𝑐𝑁))
Theorem	cxpcom 26254	Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴↑𝑐𝐵)↑𝑐𝐶) = ((𝐴↑𝑐𝐶)↑𝑐𝐵))
Theorem	dvcxp1 26255*	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+ ↦ (𝑥↑𝑐𝐴))) = (𝑥 ∈ ℝ+ ↦ (𝐴 · (𝑥↑𝑐(𝐴 − 1)))))
Theorem	dvcxp2 26256*	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 ↦ (𝐴↑𝑐𝑥))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · (𝐴↑𝑐𝑥))))
Theorem	dvsqrt 26257	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))))
Theorem	dvcncxp1 26258*	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 (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐𝐴))) = (𝑥 ∈ 𝐷 ↦ (𝐴 · (𝑥↑𝑐(𝐴 − 1)))))
Theorem	dvcnsqrt 26259*	Derivative of square root function. (Contributed by Brendan Leahy, 18-Dec-2018.)
⊢ 𝐷 = (ℂ ∖ (-∞(,]0))    ⇒   ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (1 / (2 · (√‘𝑥))))
Theorem	cxpcn 26260*	Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.)
⊢ 𝐷 = (ℂ ∖ (-∞(,]0))    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝐷)    ⇒   ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)
Theorem	cxpcn2 26261*	Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t ℝ+)    ⇒   ⊢ (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)
Theorem	cxpcn3lem 26262*	Lemma for cxpcn3 26263. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ 𝐷 = (◡ℜ “ ℝ+)    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t (0[,)+∞))    &   ⊢ 𝐿 = (𝐽 ↾t 𝐷)    &   ⊢ 𝑈 = (if((ℜ‘𝐴) ≤ 1, (ℜ‘𝐴), 1) / 2)    &   ⊢ 𝑇 = if(𝑈 ≤ (𝐸↑𝑐(1 / 𝑈)), 𝑈, (𝐸↑𝑐(1 / 𝑈)))    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ ℝ+) → ∃𝑑 ∈ ℝ+ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((abs‘𝑎) < 𝑑 ∧ (abs‘(𝐴 − 𝑏)) < 𝑑) → (abs‘(𝑎↑𝑐𝑏)) < 𝐸))
Theorem	cxpcn3 26263*	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 = (𝐽 ↾t (0[,)+∞))    &   ⊢ 𝐿 = (𝐽 ↾t 𝐷)    ⇒   ⊢ (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)
Theorem	resqrtcn 26264	Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
Theorem	sqrtcn 26265	Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ 𝐷 = (ℂ ∖ (-∞(,]0))    ⇒   ⊢ (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ)
Theorem	cxpaddlelem 26266	Lemma for cxpaddle 26267. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 1)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ≤ 1)    ⇒   ⊢ (𝜑 → 𝐴 ≤ (𝐴↑𝑐𝐵))
Theorem	cxpaddle 26267	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ≤ 1)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)))
Theorem	abscxpbnd 26268	Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵))    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑀)    ⇒   ⊢ (𝜑 → (abs‘(𝐴↑𝑐𝐵)) ≤ ((𝑀↑𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
Theorem	root1id 26269	Property of an 𝑁-th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑁 ∈ ℕ → ((-1↑𝑐(2 / 𝑁))↑𝑁) = 1)
Theorem	root1eq1 26270	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 ↔ 𝑁 ∥ 𝐾))
Theorem	root1cj 26271	Within the 𝑁-th roots of unity, the conjugate of the 𝐾-th root is the 𝑁 − 𝐾-th root. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) = ((-1↑𝑐(2 / 𝑁))↑(𝑁 − 𝐾)))
Theorem	cxpeq 26272*	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) → ((𝐴↑𝑁) = 𝐵 ↔ ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑𝑐(1 / 𝑁)) · ((-1↑𝑐(2 / 𝑁))↑𝑛))))
Theorem	loglesqrt 26273	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))
Theorem	logreclem 26274	Symmetry of the natural logarithm range by negation. Lemma for logrec 26275. (Contributed by Saveliy Skresanov, 27-Dec-2016.)
⊢ ((𝐴 ∈ ran log ∧ ¬ (ℑ‘𝐴) = π) → -𝐴 ∈ ran log)
Theorem	logrec 26275	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 ℂ, 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 (( logb ‘𝐵)‘𝑋); that looks a little more like traditional notation. Such a function ( logb ‘𝐵) for a fixed base can be obtained in Metamath (without the need for a new definition) by the curry function: (curry logb ‘𝐵), see logbmpt 26300, logbf 26301 and logbfval 26302.
Syntax	clogb 26276	Extend class notation to include the logarithm generalized to an arbitrary base.
class logb
Definition	df-logb 26277*	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)))
Theorem	logbval 26278	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)))
Theorem	logbcl 26279	General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) ∈ ℂ)
Theorem	logbid1 26280	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)
Theorem	logb1 26281	The logarithm of 1 to an arbitrary base 𝐵 is 0. Property 1(b) of [Cohen4] p. 361. See log1 26101. (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)
Theorem	elogb 26282	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))
Theorem	logbchbase 26283	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)))
Theorem	relogbval 26284	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)))
Theorem	relogbcl 26285	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)
Theorem	relogbzcl 26286	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)
Theorem	relogbreexp 26287	Power law for the general logarithm for real powers: The logarithm of a positive real number to the power of a real number is equal to the product of the exponent and the logarithm of the base of the power. Property 4 of [Cohen4] p. 361. (Contributed by AV, 9-Jun-2020.)
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ) → (𝐵 logb (𝐶↑𝑐𝐸)) = (𝐸 · (𝐵 logb 𝐶)))
Theorem	relogbzexp 26288	Power law for the general logarithm for integer powers: The logarithm of a positive real number to the power of an integer is equal to the product of the exponent and the logarithm of the base of the power. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐶 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐵 logb (𝐶↑𝑁)) = (𝑁 · (𝐵 logb 𝐶)))
Theorem	relogbmul 26289	The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.)
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 · 𝐶)) = ((𝐵 logb 𝐴) + (𝐵 logb 𝐶)))
Theorem	relogbmulexp 26290	The logarithm of the product of a positive real and a positive real number to the power of a real number is the sum of the logarithm of the first real number and the scaled logarithm of the second real number. (Contributed by AV, 29-May-2020.)
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐶↑𝑐𝐸))) = ((𝐵 logb 𝐴) + (𝐸 · (𝐵 logb 𝐶))))
Theorem	relogbdiv 26291	The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.)
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 / 𝐶)) = ((𝐵 logb 𝐴) − (𝐵 logb 𝐶)))
Theorem	relogbexp 26292	Identity law for general logarithm: the logarithm of a power to the base is the exponent. Property 6 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1 ∧ 𝑀 ∈ ℤ) → (𝐵 logb (𝐵↑𝑀)) = 𝑀)
Theorem	nnlogbexp 26293	Identity law for general logarithm with integer base. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐵 logb (𝐵↑𝑀)) = 𝑀)
Theorem	logbrec 26294	Logarithm of a reciprocal changes sign. See logrec 26275. Particular case of Property 3 of [Cohen4] p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ+) → (𝐵 logb (1 / 𝐴)) = -(𝐵 logb 𝐴))
Theorem	logbleb 26295	The general logarithm function is monotone/increasing. See logleb 26118. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+ ∧ 𝑌 ∈ ℝ+) → (𝑋 ≤ 𝑌 ↔ (𝐵 logb 𝑋) ≤ (𝐵 logb 𝑌)))
Theorem	logblt 26296	The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 26115. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+ ∧ 𝑌 ∈ ℝ+) → (𝑋 < 𝑌 ↔ (𝐵 logb 𝑋) < (𝐵 logb 𝑌)))
Theorem	relogbcxp 26297	Identity law for the general logarithm for real numbers. (Contributed by AV, 22-May-2020.)
⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ 𝑋 ∈ ℝ) → (𝐵 logb (𝐵↑𝑐𝑋)) = 𝑋)
Theorem	cxplogb 26298	Identity law for the general logarithm. (Contributed by AV, 22-May-2020.)
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋)
Theorem	relogbcxpb 26299	The logarithm is the inverse of the exponentiation. Observation in [Cohen4] p. 348. (Contributed by AV, 11-Jun-2020.)
⊢ (((𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ ℝ+ ∧ 𝑌 ∈ ℝ) → ((𝐵 logb 𝑋) = 𝑌 ↔ (𝐵↑𝑐𝑌) = 𝑋))
Theorem	logbmpt 26300*	The general logarithm to a fixed base regarded as mapping. (Contributed by AV, 11-Jun-2020.)
⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))