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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cxpsubd 25301 | 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 25302 | 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 25303 | 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 25304 | 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 25305 | 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 25306 | 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 25307 | 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 25308 | 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 25309 | 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 25310 | 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 25311 | 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 | cxpsqrtth 25312 | Square root theorem over the complex numbers for the complex power function. Theorem I.35 of [Apostol] p. 29. Compare with sqrtth 14724. (Contributed by AV, 23-Dec-2022.) |
⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑𝑐2) = 𝐴) | ||
Theorem | 2irrexpq 25313* | 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 15602), 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 25378. (Contributed by AV, 23-Dec-2022.) |
⊢ ∃𝑎 ∈ (ℝ ∖ ℚ)∃𝑏 ∈ (ℝ ∖ ℚ)(𝑎↑𝑐𝑏) ∈ ℚ | ||
Theorem | cxprecd 25314 | 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 25315 | 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 25316 | 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 25317 | 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 25318 | 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 25319 | 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 | cxpcom 25320 | 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 25321* | 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 25322* | 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 25323 | 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 25324* | 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 25325* | 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 25326* | 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 25327* | 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 25328* | Lemma for cxpcn3 25329. (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 25329* | 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 25330 | Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ) | ||
Theorem | sqrtcn 25331 | Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) ⇒ ⊢ (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ) | ||
Theorem | cxpaddlelem 25332 | Lemma for cxpaddle 25333. (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 25333 | 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 25334 | 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 25335 | 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 25336 | 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 25337 | 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 25338* | 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 25339 | 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 25340 | Symmetry of the natural logarithm range by negation. Lemma for logrec 25341. (Contributed by Saveliy Skresanov, 27-Dec-2016.) |
⊢ ((𝐴 ∈ ran log ∧ ¬ (ℑ‘𝐴) = π) → -𝐴 ∈ ran log) | ||
Theorem | logrec 25341 | 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))) | ||
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 25366, logbf 25367 and logbfval 25368. | ||
Syntax | clogb 25342 | Extend class notation to include the logarithm generalized to an arbitrary base. |
class logb | ||
Definition | df-logb 25343* | 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 25344 | 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 25345 | General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.) |
⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) ∈ ℂ) | ||
Theorem | logbid1 25346 | 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 25347 | The logarithm of 1 to an arbitrary base 𝐵 is 0. Property 1(b) of [Cohen4] p. 361. See log1 25169. (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 25348 | 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 25349 | 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 25350 | 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 25351 | 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 25352 | 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 25353 | 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 25354 | 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 25355 | 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 25356 | 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 25357 | 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 25358 | 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 25359 | 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 25360 | Logarithm of a reciprocal changes sign. See logrec 25341. 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 25361 | The general logarithm function is monotone/increasing. See logleb 25186. (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 25362 | The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 25183. (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 25363 | 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 25364 | 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 25365 | 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 25366* | 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)))) | ||
Theorem | logbf 25367 | The general logarithm to a fixed base regarded as function. (Contributed by AV, 11-Jun-2020.) |
⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵):(ℂ ∖ {0})⟶ℂ) | ||
Theorem | logbfval 25368 | 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 logb ‘𝐵)‘𝑋) = (𝐵 logb 𝑋)) | ||
Theorem | relogbf 25369 | 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 logb ‘𝐵) ↾ ℝ+):ℝ+⟶ℝ) | ||
Theorem | logblog 25370 | 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 | ||
Theorem | logbgt0b 25371 | 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)) | ||
Theorem | logbgcd1irr 25372 | 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 25373). (Contributed by AV, 29-Dec-2022.) |
⊢ ((𝑋 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2) ∧ (𝑋 gcd 𝐵) = 1) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ)) | ||
Theorem | 2logb9irr 25373 | Example for logbgcd1irr 25372. The logarithm of nine to base two is irrational. (Contributed by AV, 29-Dec-2022.) |
⊢ (2 logb 9) ∈ (ℝ ∖ ℚ) | ||
Theorem | logbprmirr 25374 | The logarithm of a prime to a different prime base is an irrational number. For example, (2 logb 3) ∈ (ℝ ∖ ℚ) (see 2logb3irr 25375). (Contributed by AV, 31-Dec-2022.) |
⊢ ((𝑋 ∈ ℙ ∧ 𝐵 ∈ ℙ ∧ 𝑋 ≠ 𝐵) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ)) | ||
Theorem | 2logb3irr 25375 | Example for logbprmirr 25374. The logarithm of three to base two is irrational. (Contributed by AV, 31-Dec-2022.) |
⊢ (2 logb 3) ∈ (ℝ ∖ ℚ) | ||
Theorem | 2logb9irrALT 25376 | Alternate proof of 2logb9irr 25373: 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) | ||
Theorem | sqrt2cxp2logb9e3 25377 | 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 15604 resp. 2logb9irr 25373), satisfying the statement in 2irrexpqALT 25378. (Contributed by AV, 29-Dec-2022.) |
⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 | ||
Theorem | 2irrexpqALT 25378* | Alternate proof of 2irrexpq 25313: There exist irrational numbers 𝑎 and 𝑏 such that (𝑎↑𝑏) 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 25313, this is a constructive proof because it is based on two explicitly named irrational numbers (√‘2) and (2 logb 9), see sqrt2irr0 15604, 2logb9irr 25373 and sqrt2cxp2logb9e3 25377. 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)(𝑎↑𝑐𝑏) ∈ ℚ | ||
Theorem | angval 25379* | 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 ((𝐶 − 𝐵)𝐹(𝐴 − 𝐵)). (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)))) | ||
Theorem | angcan 25380* | 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)) | ||
Theorem | angneg 25381* | 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)) | ||
Theorem | angvald 25382* | The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 25379. (Contributed by David Moews, 28-Feb-2017.) |
⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ≠ 0) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ≠ 0) ⇒ ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) | ||
Theorem | angcld 25383* | 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)))) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ≠ 0) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ≠ 0) ⇒ ⊢ (𝜑 → (𝑋𝐹𝑌) ∈ (-π(,]π)) | ||
Theorem | angrteqvd 25384* | 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)))) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ≠ 0) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ≠ 0) ⇒ ⊢ (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (ℜ‘(𝑌 / 𝑋)) = 0)) | ||
Theorem | cosangneg2d 25385* | 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)))) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ≠ 0) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ≠ 0) ⇒ ⊢ (𝜑 → (cos‘(𝑋𝐹-𝑌)) = -(cos‘(𝑋𝐹𝑌))) | ||
Theorem | angrtmuld 25386* | 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)))) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → 𝑍 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ≠ 0) & ⊢ (𝜑 → 𝑌 ≠ 0) & ⊢ (𝜑 → 𝑍 ≠ 0) & ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)})) | ||
Theorem | ang180lem1 25387* | Lemma for ang180 25392. Show that the "revolution number" 𝑁 is an integer, using efeq1 25113 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)) | ||
Theorem | ang180lem2 25388* | Lemma for ang180 25392. 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 25153, 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 < 𝑁 ∧ 𝑁 < 1)) | ||
Theorem | ang180lem3 25389* | Lemma for ang180 25392. Since ang180lem1 25387 shows that 𝑁 is an integer and ang180lem2 25388 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)}) | ||
Theorem | ang180lem4 25390* | Lemma for ang180 25392. 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}) | ||
Theorem | ang180lem5 25391* | Lemma for ang180 25392: 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) ∧ 𝐴 ≠ 𝐵) → ((((𝐴 − 𝐵)𝐹𝐴) + (𝐵𝐹(𝐵 − 𝐴))) + (𝐴𝐹𝐵)) ∈ {-π, π}) | ||
Theorem | ang180 25392* | The sum of angles 𝑚𝐴𝐵𝐶 + 𝑚𝐵𝐶𝐴 + 𝑚𝐶𝐴𝐵 in a triangle adds up to either π or -π, 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) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶)) → ((((𝐶 − 𝐵)𝐹(𝐴 − 𝐵)) + ((𝐴 − 𝐶)𝐹(𝐵 − 𝐶))) + ((𝐵 − 𝐴)𝐹(𝐶 − 𝐴))) ∈ {-π, π}) | ||
Theorem | lawcoslem1 25393 | Lemma for lawcos 25394. 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.) |
⊢ (𝜑 → 𝑈 ∈ ℂ) & ⊢ (𝜑 → 𝑉 ∈ ℂ) & ⊢ (𝜑 → 𝑈 ≠ 0) & ⊢ (𝜑 → 𝑉 ≠ 0) ⇒ ⊢ (𝜑 → ((abs‘(𝑈 − 𝑉))↑2) = ((((abs‘𝑈)↑2) + ((abs‘𝑉)↑2)) − (2 · (((abs‘𝑈) · (abs‘𝑉)) · ((ℜ‘(𝑈 / 𝑉)) / (abs‘(𝑈 / 𝑉))))))) | ||
Theorem | lawcos 25394* | 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 25392), 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 25393 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 15500). The Pythagorean theorem pythag 25395 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‘(𝐵 − 𝐶)) & ⊢ 𝑌 = (abs‘(𝐴 − 𝐶)) & ⊢ 𝑍 = (abs‘(𝐴 − 𝐵)) & ⊢ 𝑂 = ((𝐵 − 𝐶)𝐹(𝐴 − 𝐶)) ⇒ ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝑍↑2) = (((𝑋↑2) + (𝑌↑2)) − (2 · ((𝑋 · 𝑌) · (cos‘𝑂))))) | ||
Theorem | pythag 25395* | 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 25392), 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 25394 to prove this, along with simple trigonometry facts like coshalfpi 25055 and cosneg 15500. (Contributed by David A. Wheeler, 13-Jun-2015.) |
⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) & ⊢ 𝑋 = (abs‘(𝐵 − 𝐶)) & ⊢ 𝑌 = (abs‘(𝐴 − 𝐶)) & ⊢ 𝑍 = (abs‘(𝐴 − 𝐵)) & ⊢ 𝑂 = ((𝐵 − 𝐶)𝐹(𝐴 − 𝐶)) ⇒ ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑂 ∈ {(π / 2), -(π / 2)}) → (𝑍↑2) = ((𝑋↑2) + (𝑌↑2))) | ||
Theorem | isosctrlem1 25396 | Lemma for isosctr 25399. (Contributed by Saveliy Skresanov, 30-Dec-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π) | ||
Theorem | isosctrlem2 25397 | Lemma for isosctr 25399. 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))))) | ||
Theorem | isosctrlem3 25398* | Lemma for isosctr 25399. 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 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘𝐴) = (abs‘𝐵)) → (-𝐴𝐹(𝐵 − 𝐴)) = ((𝐴 − 𝐵)𝐹-𝐵)) | ||
Theorem | isosctr 25399* | 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) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → ((𝐶 − 𝐴)𝐹(𝐵 − 𝐴)) = ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵))) | ||
Theorem | ssscongptld 25400* |
If two triangles have equal sides, one angle in one triangle has the
same cosine as the corresponding angle in the other triangle. This is a
partial form of the SSS congruence theorem.
This theorem is proven by using lawcos 25394 on both triangles to express one side in terms of the other two, and then equating these expressions and reducing this algebraically to get an equality of cosines of angles. (Contributed by David Moews, 28-Feb-2017.) |
⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐸 ∈ ℂ) & ⊢ (𝜑 → 𝐺 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐷 ≠ 𝐸) & ⊢ (𝜑 → 𝐸 ≠ 𝐺) & ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐷 − 𝐸))) & ⊢ (𝜑 → (abs‘(𝐵 − 𝐶)) = (abs‘(𝐸 − 𝐺))) & ⊢ (𝜑 → (abs‘(𝐶 − 𝐴)) = (abs‘(𝐺 − 𝐷))) ⇒ ⊢ (𝜑 → (cos‘((𝐴 − 𝐵)𝐹(𝐶 − 𝐵))) = (cos‘((𝐷 − 𝐸)𝐹(𝐺 − 𝐸)))) |
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