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Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcxpaddd 25001 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 25002 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 25003 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 25004 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 25005 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 25006 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 25007 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 25008 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 25009 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 25010 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 25011 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 25012 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)))

Theoremcxpsqrtth 25013 Square root theorem over the complex numbers for the complex power function. Theorem I.35 of [Apostol] p. 29. Compare with sqrtth 14585. (Contributed by AV, 23-Dec-2022.)
(𝐴 ∈ ℂ → ((√‘𝐴)↑𝑐2) = 𝐴)

Theorem2irrexpq 25014* There exist irrational numbers 𝑎 and 𝑏 such that (𝑎𝑐𝑏) 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 15462), 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 25079. (Contributed by AV, 23-Dec-2022.)
𝑎 ∈ (ℝ ∖ ℚ)∃𝑏 ∈ (ℝ ∖ ℚ)(𝑎𝑐𝑏) ∈ ℚ

Theoremcxprecd 25015 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴𝑐𝐵)))

Theoremrpcxpcld 25016 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝑐𝐵) ∈ ℝ+)

Theoremlogcxpd 25017 Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))

Theoremcxplt3d 25018 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 25019 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 25020 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))

Theoremcxpcom 25021 Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝑐𝐵)↑𝑐𝐶) = ((𝐴𝑐𝐶)↑𝑐𝐵))

Theoremdvcxp1 25022* 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 25023* 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 25024 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 25025* 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 25026* Derivative of square root function. (Contributed by Brendan Leahy, 18-Dec-2018.)
𝐷 = (ℂ ∖ (-∞(,]0))       (ℂ D (𝑥𝐷 ↦ (√‘𝑥))) = (𝑥𝐷 ↦ (1 / (2 · (√‘𝑥))))

Theoremcxpcn 25027* 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 25028* 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 25029* Lemma for cxpcn3 25030. (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 25030* 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 25031 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)
(√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)

Theoremsqrtcn 25032 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 25034 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 25035 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 25036 Property of an 𝑁-th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝑁 ∈ ℕ → ((-1↑𝑐(2 / 𝑁))↑𝑁) = 1)

Theoremroot1eq1 25037 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 25038 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 25039* 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 25040 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 25041 Symmetry of the natural logarithm range by negation. Lemma for logrec 25042. (Contributed by Saveliy Skresanov, 27-Dec-2016.)
((𝐴 ∈ ran log ∧ ¬ (ℑ‘𝐴) = π) → -𝐴 ∈ ran log)

Theoremlogrec 25042 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 25067, logbf 25068 and logbfval 25069.

Syntaxclogb 25043 Extend class notation to include the logarithm generalized to an arbitrary base.
class logb

Definitiondf-logb 25044* 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 25045 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 25046 General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) ∈ ℂ)

Theoremlogbid1 25047 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 25048 The logarithm of 1 to an arbitrary base 𝐵 is 0. Property 1(b) of [Cohen4] p. 361. See log1 24870. (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 25049 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 25050 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 25051 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 25052 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 25053 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)

Theoremrelogbreexp 25054 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 (𝐶𝑐𝐸)) = (𝐸 · (𝐵 logb 𝐶)))

Theoremrelogbzexp 25055 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 (𝐶𝑁)) = (𝑁 · (𝐵 logb 𝐶)))

Theoremrelogbmul 25056 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)))

Theoremrelogbmulexp 25057 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 · (𝐶𝑐𝐸))) = ((𝐵 logb 𝐴) + (𝐸 · (𝐵 logb 𝐶))))

Theoremrelogbdiv 25058 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)))

Theoremrelogbexp 25059 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 (𝐵𝑀)) = 𝑀)

Theoremnnlogbexp 25060 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 (𝐵𝑀)) = 𝑀)

Theoremlogbrec 25061 Logarithm of a reciprocal changes sign. See logrec 25042. 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))

Theoremlogbleb 25062 The general logarithm function is monotone/increasing. See logleb 24887. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℝ+𝑌 ∈ ℝ+) → (𝑋𝑌 ↔ (𝐵 logb 𝑋) ≤ (𝐵 logb 𝑌)))

Theoremlogblt 25063 The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 24884. (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)))

Theoremrelogbcxp 25064 Identity law for the general logarithm for real numbers. (Contributed by AV, 22-May-2020.)
((𝐵 ∈ (ℝ+ ∖ {1}) ∧ 𝑋 ∈ ℝ) → (𝐵 logb (𝐵𝑐𝑋)) = 𝑋)

Theoremcxplogb 25065 Identity law for the general logarithm. (Contributed by AV, 22-May-2020.)
((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵𝑐(𝐵 logb 𝑋)) = 𝑋)

Theoremrelogbcxpb 25066 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 ↔ (𝐵𝑐𝑌) = 𝑋))

Theoremlogbmpt 25067* The general logarithm to a fixed base regarded as mapping. (Contributed by AV, 11-Jun-2020.)
((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb𝐵) = (𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝐵))))

Theoremlogbf 25068 The general logarithm to a fixed base regarded as function. (Contributed by AV, 11-Jun-2020.)
((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb𝐵):(ℂ ∖ {0})⟶ℂ)

Theoremlogbfval 25069 The general logarithm of a complex number to a fixed base. (Contributed by AV, 11-Jun-2020.)
(((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ((curry logb𝐵)‘𝑋) = (𝐵 logb 𝑋))

Theoremrelogbf 25070 The general logarithm to a real base greater than 1 regarded as function restricted to the positive integers. Property in [Cohen4] p. 349. (Contributed by AV, 12-Jun-2020.)
((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → ((curry logb𝐵) ↾ ℝ+):ℝ+⟶ℝ)

Theoremlogblog 25071 The general logarithm to the base being Euler's constant regarded as function is the natural logarithm. (Contributed by AV, 12-Jun-2020.)
(curry logb ‘e) = log

Theoremlogbgt0b 25072 The logarithm of a positive real number to a real base greater than 1 is positive iff the number is greater than 1. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℝ+ ∧ (𝐵 ∈ ℝ+ ∧ 1 < 𝐵)) → (0 < (𝐵 logb 𝐴) ↔ 1 < 𝐴))

Theoremlogbgcd1irr 25073 The logarithm of an integer greater than 1 to an integer base greater than 1 is an irrational number if the argument and the base are relatively prime. For example, (2 logb 9) ∈ (ℝ ∖ ℚ) (see 2logb9irr 25074). (Contributed by AV, 29-Dec-2022.)
((𝑋 ∈ (ℤ‘2) ∧ 𝐵 ∈ (ℤ‘2) ∧ (𝑋 gcd 𝐵) = 1) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ))

Theorem2logb9irr 25074 Example for logbgcd1irr 25073. The logarithm of nine to base two is irrational. (Contributed by AV, 29-Dec-2022.)
(2 logb 9) ∈ (ℝ ∖ ℚ)

Theoremlogbprmirr 25075 The logarithm of a prime to a different prime base is an irrational number. For example, (2 logb 3) ∈ (ℝ ∖ ℚ) (see 2logb3irr 25076). (Contributed by AV, 31-Dec-2022.)
((𝑋 ∈ ℙ ∧ 𝐵 ∈ ℙ ∧ 𝑋𝐵) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ))

Theorem2logb3irr 25076 Example for logbprmirr 25075. The logarithm of three to base two is irrational. (Contributed by AV, 31-Dec-2022.)
(2 logb 3) ∈ (ℝ ∖ ℚ)

Theorem2logb9irrALT 25077 Alternate proof of 2logb9irr 25074: The logarithm of nine to base two is irrational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(2 logb 9) ∈ (ℝ ∖ ℚ)

Theoremsqrt2cxp2logb9e3 25078 The square root of two to the power of the logarithm of nine to base two is three. (√‘2) and (2 logb 9) are irrational numbers (see sqrt2irr0 15464 resp. 2logb9irr 25074), satisfying the statement in 2irrexpqALT 25079. (Contributed by AV, 29-Dec-2022.)
((√‘2)↑𝑐(2 logb 9)) = 3

Theorem2irrexpqALT 25079* Alternate proof of 2irrexpq 25014: There exist irrational numbers 𝑎 and 𝑏 such that (𝑎𝑏) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. In contrast to 2irrexpq 25014, this is a constructive proof because it is based on two explicitly named irrational numbers (√‘2) and (2 logb 9), see sqrt2irr0 15464, 2logb9irr 25074 and sqrt2cxp2logb9e3 25078. Therefore, this proof is also acceptable/usable in intuitionistic logic. (Contributed by AV, 23-Dec-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑎 ∈ (ℝ ∖ ℚ)∃𝑏 ∈ (ℝ ∖ ℚ)(𝑎𝑐𝑏) ∈ ℚ

14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords

Theoremangval 25080* Define the angle function, which takes two complex numbers, treated as vectors from the origin, and returns the angle between them, in the range ( − π, π]. To convert from the geometry notation, 𝑚𝐴𝐵𝐶, the measure of the angle with legs 𝐴𝐵, 𝐶𝐵 where 𝐶 is more counterclockwise for positive angles, is represented by ((𝐶𝐵)𝐹(𝐴𝐵)). (Contributed by Mario Carneiro, 23-Sep-2014.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))       (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴))))

Theoremangcan 25081* Cancel a constant multiplier in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))       (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐴)𝐹(𝐶 · 𝐵)) = (𝐴𝐹𝐵))

Theoremangneg 25082* Cancel a negative sign in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))       (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (-𝐴𝐹-𝐵) = (𝐴𝐹𝐵))

Theoremangvald 25083* The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 25080. (Contributed by David Moews, 28-Feb-2017.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑𝑌 ≠ 0)       (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))

Theoremangcld 25084* The (signed) angle between two vectors is in (-π(,]π). Deduction form. (Contributed by David Moews, 28-Feb-2017.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑𝑌 ≠ 0)       (𝜑 → (𝑋𝐹𝑌) ∈ (-π(,]π))

Theoremangrteqvd 25085* Two vectors are at a right angle iff their quotient is purely imaginary. (Contributed by David Moews, 28-Feb-2017.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑𝑌 ≠ 0)       (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (ℜ‘(𝑌 / 𝑋)) = 0))

Theoremcosangneg2d 25086* The cosine of the angle between 𝑋 and -𝑌 is the negative of that between 𝑋 and 𝑌. If A, B and C are collinear points, this implies that the cosines of DBA and DBC sum to zero, i.e., that DBA and DBC are supplementary. (Contributed by David Moews, 28-Feb-2017.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑𝑌 ≠ 0)       (𝜑 → (cos‘(𝑋𝐹-𝑌)) = -(cos‘(𝑋𝐹𝑌)))

Theoremangrtmuld 25087* Perpendicularity of two vectors does not change under rescaling the second. (Contributed by David Moews, 28-Feb-2017.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 ∈ ℂ)    &   (𝜑𝑍 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)    &   (𝜑𝑌 ≠ 0)    &   (𝜑𝑍 ≠ 0)    &   (𝜑 → (𝑍 / 𝑌) ∈ ℝ)       (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)}))

Theoremang180lem1 25088* Lemma for ang180 25093. Show that the "revolution number" 𝑁 is an integer, using efeq1 24814 to show that since the product of the three arguments 𝐴, 1 / (1 − 𝐴), (𝐴 − 1) / 𝐴 is -1, the sum of the logarithms must be an integer multiple of 2πi away from πi = log(-1). (Contributed by Mario Carneiro, 23-Sep-2014.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   𝑇 = (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))    &   𝑁 = (((𝑇 / i) / (2 · π)) − (1 / 2))       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 ∈ ℤ ∧ (𝑇 / i) ∈ ℝ))

Theoremang180lem2 25089* Lemma for ang180 25093. Show that the revolution number 𝑁 is strictly between -2 and 1. Both bounds are established by iterating using the bounds on the imaginary part of the logarithm, logimcl 24854, but the resulting bound gives only 𝑁 ≤ 1 for the upper bound. The case 𝑁 = 1 is not ruled out here, but it is in some sense an "edge case" that can only happen under very specific conditions; in particular we show that all the angle arguments 𝐴, 1 / (1 − 𝐴), (𝐴 − 1) / 𝐴 must lie on the negative real axis, which is a contradiction because clearly if 𝐴 is negative then the other two are positive real. (Contributed by Mario Carneiro, 23-Sep-2014.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   𝑇 = (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))    &   𝑁 = (((𝑇 / i) / (2 · π)) − (1 / 2))       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-2 < 𝑁𝑁 < 1))

Theoremang180lem3 25090* Lemma for ang180 25093. Since ang180lem1 25088 shows that 𝑁 is an integer and ang180lem2 25089 shows that 𝑁 is strictly between -2 and 1, it follows that 𝑁 ∈ {-1, 0}, and these two cases correspond to the two possible values for 𝑇. (Contributed by Mario Carneiro, 23-Sep-2014.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   𝑇 = (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))    &   𝑁 = (((𝑇 / i) / (2 · π)) − (1 / 2))       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑇 ∈ {-(i · π), (i · π)})

Theoremang180lem4 25091* Lemma for ang180 25093. Reduce the statement to one variable. (Contributed by Mario Carneiro, 23-Sep-2014.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((((1 − 𝐴)𝐹1) + (𝐴𝐹(𝐴 − 1))) + (1𝐹𝐴)) ∈ {-π, π})

Theoremang180lem5 25092* Lemma for ang180 25093: Reduce the statement to two variables. (Contributed by Mario Carneiro, 23-Sep-2014.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))       (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝐴𝐵) → ((((𝐴𝐵)𝐹𝐴) + (𝐵𝐹(𝐵𝐴))) + (𝐴𝐹𝐵)) ∈ {-π, π})

Theoremang180 25093* The sum of angles 𝑚𝐴𝐵𝐶 + 𝑚𝐵𝐶𝐴 + 𝑚𝐶𝐴𝐵 in a triangle adds up to either π or , i.e. 180 degrees. (The sign is due to the two possible orientations of vertex arrangement and our signed notion of angle). This is Metamath 100 proof #27. (Contributed by Mario Carneiro, 23-Sep-2014.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))       (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴𝐵𝐵𝐶𝐴𝐶)) → ((((𝐶𝐵)𝐹(𝐴𝐵)) + ((𝐴𝐶)𝐹(𝐵𝐶))) + ((𝐵𝐴)𝐹(𝐶𝐴))) ∈ {-π, π})

Theoremlawcoslem1 25094 Lemma for lawcos 25095. Here we prove the law for a point at the origin and two distinct points U and V, using an expanded version of the signed angle expression on the complex plane. (Contributed by David A. Wheeler, 11-Jun-2015.)
(𝜑𝑈 ∈ ℂ)    &   (𝜑𝑉 ∈ ℂ)    &   (𝜑𝑈 ≠ 0)    &   (𝜑𝑉 ≠ 0)       (𝜑 → ((abs‘(𝑈𝑉))↑2) = ((((abs‘𝑈)↑2) + ((abs‘𝑉)↑2)) − (2 · (((abs‘𝑈) · (abs‘𝑉)) · ((ℜ‘(𝑈 / 𝑉)) / (abs‘(𝑈 / 𝑉)))))))

Theoremlawcos 25095* Law of cosines (also known as the Al-Kashi theorem or the generalized Pythagorean theorem, or the cosine formula or cosine rule). Given three distinct points A, B, and C, prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where 𝐹 is the signed angle construct (as used in ang180 25093), 𝑋 is the distance of line segment BC, 𝑌 is the distance of line segment AC, 𝑍 is the distance of line segment AB, and 𝑂 is the signed angle m/__ BCA on the complex plane. We translate triangle ABC to move C to the origin (C-C), B to U=(B-C), and A to V=(A-C), then use lemma lawcoslem1 25094 to prove this algebraically simpler case. The Metamath convention is to use a signed angle; in this case the sign doesn't matter because we use the cosine of the angle (see cosneg 15360). The Pythagorean theorem pythag 25096 is a special case of the law of cosines. The theorem's expression and approach were suggested by Mario Carneiro. This is Metamath 100 proof #94. (Contributed by David A. Wheeler, 12-Jun-2015.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   𝑋 = (abs‘(𝐵𝐶))    &   𝑌 = (abs‘(𝐴𝐶))    &   𝑍 = (abs‘(𝐴𝐵))    &   𝑂 = ((𝐵𝐶)𝐹(𝐴𝐶))       (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴𝐶𝐵𝐶)) → (𝑍↑2) = (((𝑋↑2) + (𝑌↑2)) − (2 · ((𝑋 · 𝑌) · (cos‘𝑂)))))

Theorempythag 25096* Pythagorean theorem. Given three distinct points A, B, and C that form a right triangle (with the right angle at C), prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where 𝐹 is the signed angle construct (as used in ang180 25093), 𝑋 is the distance of line segment BC, 𝑌 is the distance of line segment AC, 𝑍 is the distance of line segment AB (the hypotenuse), and 𝑂 is the signed right angle m/__ BCA. We use the law of cosines lawcos 25095 to prove this, along with simple trigonometry facts like coshalfpi 24758 and cosneg 15360. (Contributed by David A. Wheeler, 13-Jun-2015.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   𝑋 = (abs‘(𝐵𝐶))    &   𝑌 = (abs‘(𝐴𝐶))    &   𝑍 = (abs‘(𝐴𝐵))    &   𝑂 = ((𝐵𝐶)𝐹(𝐴𝐶))       (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝑂 ∈ {(π / 2), -(π / 2)}) → (𝑍↑2) = ((𝑋↑2) + (𝑌↑2)))

Theoremisosctrlem1 25097 Lemma for isosctr 25100. (Contributed by Saveliy Skresanov, 30-Dec-2016.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)

Theoremisosctrlem2 25098 Lemma for isosctr 25100. Corresponds to the case where one vertex is at 0, another at 1 and the third lies on the unit circle. (Contributed by Saveliy Skresanov, 31-Dec-2016.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))

Theoremisosctrlem3 25099* Lemma for isosctr 25100. Corresponds to the case where one vertex is at 0. (Contributed by Saveliy Skresanov, 1-Jan-2017.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))       (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ∧ 𝐴𝐵) ∧ (abs‘𝐴) = (abs‘𝐵)) → (-𝐴𝐹(𝐵𝐴)) = ((𝐴𝐵)𝐹-𝐵))

Theoremisosctr 25100* Isosceles triangle theorem. This is Metamath 100 proof #65. (Contributed by Saveliy Skresanov, 1-Jan-2017.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))       (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴𝐶𝐵𝐶𝐴𝐵) ∧ (abs‘(𝐴𝐶)) = (abs‘(𝐵𝐶))) → ((𝐶𝐴)𝐹(𝐵𝐴)) = ((𝐴𝐵)𝐹(𝐶𝐵)))

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