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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cos2pim 25001 | Cosine of a number subtracted from 2 · π. (Contributed by Paul Chapman, 15-Mar-2008.) |
⊢ (𝐴 ∈ ℂ → (cos‘((2 · π) − 𝐴)) = (cos‘𝐴)) | ||
Theorem | sinmpi 25002 | Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.) |
⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴)) | ||
Theorem | cosmpi 25003 | Cosine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.) |
⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 − π)) = -(cos‘𝐴)) | ||
Theorem | sinppi 25004 | Sine of a number plus π. (Contributed by NM, 10-Aug-2008.) |
⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 + π)) = -(sin‘𝐴)) | ||
Theorem | cosppi 25005 | Cosine of a number plus π. (Contributed by NM, 18-Aug-2008.) |
⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴)) | ||
Theorem | efimpi 25006 | The exponential function at i times a real number less π. (Contributed by Paul Chapman, 15-Mar-2008.) |
⊢ (𝐴 ∈ ℂ → (exp‘(i · (𝐴 − π))) = -(exp‘(i · 𝐴))) | ||
Theorem | sinhalfpip 25007 | The sine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴)) | ||
Theorem | sinhalfpim 25008 | The sine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴)) | ||
Theorem | coshalfpip 25009 | The cosine of π / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (cos‘((π / 2) + 𝐴)) = -(sin‘𝐴)) | ||
Theorem | coshalfpim 25010 | The cosine of π / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ ℂ → (cos‘((π / 2) − 𝐴)) = (sin‘𝐴)) | ||
Theorem | ptolemy 25011 | Ptolemy's Theorem. This theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). This particular version is expressed using the sine function. It is proved by expanding all the multiplication of sines to a product of cosines of differences using sinmul 15515, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. This is Metamath 100 proof #95. (Contributed by David A. Wheeler, 31-May-2015.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = π) → (((sin‘𝐴) · (sin‘𝐵)) + ((sin‘𝐶) · (sin‘𝐷))) = ((sin‘(𝐵 + 𝐶)) · (sin‘(𝐴 + 𝐶)))) | ||
Theorem | sincosq1lem 25012 | Lemma for sincosq1sgn 25013. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴)) | ||
Theorem | sincosq1sgn 25013 | The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴))) | ||
Theorem | sincosq2sgn 25014 | The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ ((π / 2)(,)π) → (0 < (sin‘𝐴) ∧ (cos‘𝐴) < 0)) | ||
Theorem | sincosq3sgn 25015 | The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ (π(,)(3 · (π / 2))) → ((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0)) | ||
Theorem | sincosq4sgn 25016 | The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.) |
⊢ (𝐴 ∈ ((3 · (π / 2))(,)(2 · π)) → ((sin‘𝐴) < 0 ∧ 0 < (cos‘𝐴))) | ||
Theorem | coseq00topi 25017 | Location of the zeroes of cosine in (0[,]π). (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝐴 ∈ (0[,]π) → ((cos‘𝐴) = 0 ↔ 𝐴 = (π / 2))) | ||
Theorem | coseq0negpitopi 25018 | Location of the zeroes of cosine in (-π(,]π). (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝐴 ∈ (-π(,]π) → ((cos‘𝐴) = 0 ↔ 𝐴 ∈ {(π / 2), -(π / 2)})) | ||
Theorem | tanrpcl 25019 | Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.) |
⊢ (𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+) | ||
Theorem | tangtx 25020 | The tangent function is greater than its argument on positive reals in its principal domain. (Contributed by Mario Carneiro, 29-Jul-2014.) |
⊢ (𝐴 ∈ (0(,)(π / 2)) → 𝐴 < (tan‘𝐴)) | ||
Theorem | tanabsge 25021 | The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.) |
⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴))) | ||
Theorem | sinq12gt0 25022 | The sine of a number strictly between 0 and π is positive. (Contributed by Paul Chapman, 15-Mar-2008.) |
⊢ (𝐴 ∈ (0(,)π) → 0 < (sin‘𝐴)) | ||
Theorem | sinq12ge0 25023 | The sine of a number between 0 and π is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.) |
⊢ (𝐴 ∈ (0[,]π) → 0 ≤ (sin‘𝐴)) | ||
Theorem | sinq34lt0t 25024 | The sine of a number strictly between π and 2 · π is negative. (Contributed by NM, 17-Aug-2008.) |
⊢ (𝐴 ∈ (π(,)(2 · π)) → (sin‘𝐴) < 0) | ||
Theorem | cosq14gt0 25025 | The cosine of a number strictly between -π / 2 and π / 2 is positive. (Contributed by Mario Carneiro, 25-Feb-2015.) |
⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (cos‘𝐴)) | ||
Theorem | cosq14ge0 25026 | The cosine of a number between -π / 2 and π / 2 is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.) |
⊢ (𝐴 ∈ (-(π / 2)[,](π / 2)) → 0 ≤ (cos‘𝐴)) | ||
Theorem | sincosq1eq 25027 | Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → (sin‘(𝐴 · (π / 2))) = (cos‘(𝐵 · (π / 2)))) | ||
Theorem | sincos4thpi 25028 | The sine and cosine of π / 4. (Contributed by Paul Chapman, 25-Jan-2008.) |
⊢ ((sin‘(π / 4)) = (1 / (√‘2)) ∧ (cos‘(π / 4)) = (1 / (√‘2))) | ||
Theorem | tan4thpi 25029 | The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.) |
⊢ (tan‘(π / 4)) = 1 | ||
Theorem | sincos6thpi 25030 | The sine and cosine of π / 6. (Contributed by Paul Chapman, 25-Jan-2008.) Replace OLD theorem. (Revised by Wolf Lammen, 24-Sep-2020.) |
⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2)) | ||
Theorem | sincos3rdpi 25031 | The sine and cosine of π / 3. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ ((sin‘(π / 3)) = ((√‘3) / 2) ∧ (cos‘(π / 3)) = (1 / 2)) | ||
Theorem | pigt3 25032 | π is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.) |
⊢ 3 < π | ||
Theorem | pige3 25033 | π is greater than or equal to 3. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ 3 ≤ π | ||
Theorem | pige3ALT 25034 | Alternate proof of pige3 25033. This proof is based on the geometric observation that a hexagon of unit side length has perimeter 6, which is less than the unit-radius circumcircle, of perimeter 2π. We translate this to algebra by looking at the function e↑(i𝑥) as 𝑥 goes from 0 to π / 3; it moves at unit speed and travels distance 1, hence 1 ≤ π / 3. (Contributed by Mario Carneiro, 21-May-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 3 ≤ π | ||
Theorem | abssinper 25035 | The absolute value of sine has period π. (Contributed by NM, 17-Aug-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ) → (abs‘(sin‘(𝐴 + (𝐾 · π)))) = (abs‘(sin‘𝐴))) | ||
Theorem | sinkpi 25036 | The sine of an integer multiple of π is 0. (Contributed by NM, 11-Aug-2008.) |
⊢ (𝐾 ∈ ℤ → (sin‘(𝐾 · π)) = 0) | ||
Theorem | coskpi 25037 | The absolute value of the cosine of an integer multiple of π is 1. (Contributed by NM, 19-Aug-2008.) |
⊢ (𝐾 ∈ ℤ → (abs‘(cos‘(𝐾 · π))) = 1) | ||
Theorem | sineq0 25038 | A complex number whose sine is zero is an integer multiple of π. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.) |
⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) = 0 ↔ (𝐴 / π) ∈ ℤ)) | ||
Theorem | coseq1 25039 | A complex number whose cosine is one is an integer multiple of 2π. (Contributed by Mario Carneiro, 12-May-2014.) |
⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) = 1 ↔ (𝐴 / (2 · π)) ∈ ℤ)) | ||
Theorem | efeq1 25040 | A complex number whose exponential is one is an integer multiple of 2πi. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.) |
⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) = 1 ↔ (𝐴 / (i · (2 · π))) ∈ ℤ)) | ||
Theorem | cosne0 25041 | The cosine function has no zeroes within the vertical strip of the complex plane between real part -π / 2 and π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) ≠ 0) | ||
Theorem | cosordlem 25042 | Lemma for cosord 25043. (Contributed by Mario Carneiro, 10-May-2014.) |
⊢ (𝜑 → 𝐴 ∈ (0[,]π)) & ⊢ (𝜑 → 𝐵 ∈ (0[,]π)) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (cos‘𝐵) < (cos‘𝐴)) | ||
Theorem | cosord 25043 | Cosine is decreasing over the closed interval from 0 to π. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.) |
⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 < 𝐵 ↔ (cos‘𝐵) < (cos‘𝐴))) | ||
Theorem | cos11 25044 | Cosine is one-to-one over the closed interval from 0 to π. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.) |
⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 = 𝐵 ↔ (cos‘𝐴) = (cos‘𝐵))) | ||
Theorem | sinord 25045 | Sine is increasing over the closed interval from -(π / 2) to (π / 2). (Contributed by Mario Carneiro, 29-Jul-2014.) |
⊢ ((𝐴 ∈ (-(π / 2)[,](π / 2)) ∧ 𝐵 ∈ (-(π / 2)[,](π / 2))) → (𝐴 < 𝐵 ↔ (sin‘𝐴) < (sin‘𝐵))) | ||
Theorem | recosf1o 25046 | The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) |
⊢ (cos ↾ (0[,]π)):(0[,]π)–1-1-onto→(-1[,]1) | ||
Theorem | resinf1o 25047 | The sine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) |
⊢ (sin ↾ (-(π / 2)[,](π / 2))):(-(π / 2)[,](π / 2))–1-1-onto→(-1[,]1) | ||
Theorem | tanord1 25048 | The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 25049.) (Contributed by Mario Carneiro, 29-Jul-2014.) Revised to replace an OLD theorem. (Revised by Wolf Lammen, 20-Sep-2020.) |
⊢ ((𝐴 ∈ (0[,)(π / 2)) ∧ 𝐵 ∈ (0[,)(π / 2))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵))) | ||
Theorem | tanord 25049 | The tangent function is strictly increasing on its principal domain. (Contributed by Mario Carneiro, 4-Apr-2015.) |
⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵))) | ||
Theorem | tanregt0 25050 | The real part of the tangent of a complex number with real part in the open interval (0(,)(π / 2)) is positive. (Contributed by Mario Carneiro, 5-Apr-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (0(,)(π / 2))) → 0 < (ℜ‘(tan‘𝐴))) | ||
Theorem | negpitopissre 25051 | The interval (-π(,]π) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (-π(,]π) ⊆ ℝ | ||
Theorem | efgh 25052* | The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 11-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (exp‘(𝐴 · 𝑥))) ⇒ ⊢ (((𝐴 ∈ ℂ ∧ 𝑋 ∈ (SubGrp‘ℂfld)) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐹‘(𝐵 + 𝐶)) = ((𝐹‘𝐵) · (𝐹‘𝐶))) | ||
Theorem | efif1olem1 25053* | Lemma for efif1o 25057. (Contributed by Mario Carneiro, 13-May-2014.) |
⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π))) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π)) | ||
Theorem | efif1olem2 25054* | Lemma for efif1o 25057. (Contributed by Mario Carneiro, 13-May-2014.) |
⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π))) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ) | ||
Theorem | efif1olem3 25055* | Lemma for efif1o 25057. (Contributed by Mario Carneiro, 8-May-2015.) |
⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) & ⊢ 𝐶 = (◡abs “ {1}) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (ℑ‘(√‘𝑥)) ∈ (-1[,]1)) | ||
Theorem | efif1olem4 25056* | The exponential function of an imaginary number maps any interval of length 2π one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.) |
⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) & ⊢ 𝐶 = (◡abs “ {1}) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π)) & ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ) & ⊢ 𝑆 = (sin ↾ (-(π / 2)[,](π / 2))) ⇒ ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝐶) | ||
Theorem | efif1o 25057* | The exponential function of an imaginary number maps any open-below, closed-above interval of length 2π one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.) |
⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) & ⊢ 𝐶 = (◡abs “ {1}) & ⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π))) ⇒ ⊢ (𝐴 ∈ ℝ → 𝐹:𝐷–1-1-onto→𝐶) | ||
Theorem | efifo 25058* | The exponential function of an imaginary number maps the reals onto the unit circle. (Contributed by Mario Carneiro, 13-May-2014.) |
⊢ 𝐹 = (𝑧 ∈ ℝ ↦ (exp‘(i · 𝑧))) & ⊢ 𝐶 = (◡abs “ {1}) ⇒ ⊢ 𝐹:ℝ–onto→𝐶 | ||
Theorem | eff1olem 25059* | The exponential function maps the set 𝑆, of complex numbers with imaginary part in a real interval of length 2 · π, one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.) |
⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) & ⊢ 𝑆 = (◡ℑ “ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π)) & ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ) ⇒ ⊢ (𝜑 → (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0})) | ||
Theorem | eff1o 25060 | The exponential function maps the set 𝑆, of complex numbers with imaginary part in the closed-above, open-below interval from -π to π one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.) |
⊢ 𝑆 = (◡ℑ “ (-π(,]π)) ⇒ ⊢ (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}) | ||
Theorem | efabl 25061* | The image of a subgroup of the group +, under the exponential function of a scaled complex number, is an Abelian group. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (exp‘(𝐴 · 𝑥))) & ⊢ 𝐺 = ((mulGrp‘ℂfld) ↾s ran 𝐹) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘ℂfld)) ⇒ ⊢ (𝜑 → 𝐺 ∈ Abel) | ||
Theorem | efsubm 25062* | The image of a subgroup of the group +, under the exponential function of a scaled complex number is a submonoid of the multiplicative group of ℂfld. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (exp‘(𝐴 · 𝑥))) & ⊢ 𝐺 = ((mulGrp‘ℂfld) ↾s ran 𝐹) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘ℂfld)) ⇒ ⊢ (𝜑 → ran 𝐹 ∈ (SubMnd‘(mulGrp‘ℂfld))) | ||
Theorem | circgrp 25063 | The circle group 𝑇 is an Abelian group. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.) |
⊢ 𝐶 = (◡abs “ {1}) & ⊢ 𝑇 = ((mulGrp‘ℂfld) ↾s 𝐶) ⇒ ⊢ 𝑇 ∈ Abel | ||
Theorem | circsubm 25064 | The circle group 𝑇 is a submonoid of the multiplicative group of ℂfld. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
⊢ 𝐶 = (◡abs “ {1}) & ⊢ 𝑇 = ((mulGrp‘ℂfld) ↾s 𝐶) ⇒ ⊢ 𝐶 ∈ (SubMnd‘(mulGrp‘ℂfld)) | ||
Syntax | clog 25065 | Extend class notation with the natural logarithm function on complex numbers. |
class log | ||
Syntax | ccxp 25066 | Extend class notation with the complex power function. |
class ↑𝑐 | ||
Definition | df-log 25067 | Define the natural logarithm function on complex numbers. It is defined as the principal value, that is, the inverse of the exponential whose imaginary part lies in the interval (-pi, pi]. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ log = ◡(exp ↾ (◡ℑ “ (-π(,]π))) | ||
Definition | df-cxp 25068* | Define the power function on complex numbers. Note that the value of this function when 𝑥 = 0 and (ℜ‘𝑦) ≤ 0, 𝑦 ≠ 0 should properly be undefined, but defining it by convention this way simplifies the domain. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥))))) | ||
Theorem | logrn 25069 | The range of the natural logarithm function, also the principal domain of the exponential function. This allows us to write the longer class expression as simply ran log. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.) |
⊢ ran log = (◡ℑ “ (-π(,]π)) | ||
Theorem | ellogrn 25070 | Write out the property 𝐴 ∈ ran log explicitly. (Contributed by Mario Carneiro, 1-Apr-2015.) |
⊢ (𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)) | ||
Theorem | dflog2 25071 | The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ log = ◡(exp ↾ ran log) | ||
Theorem | relogrn 25072 | The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 1-Apr-2015.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ran log) | ||
Theorem | logrncn 25073 | The range of the natural logarithm function is a subset of the complex numbers. (Contributed by Mario Carneiro, 13-May-2014.) |
⊢ (𝐴 ∈ ran log → 𝐴 ∈ ℂ) | ||
Theorem | eff1o2 25074 | The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.) |
⊢ (exp ↾ ran log):ran log–1-1-onto→(ℂ ∖ {0}) | ||
Theorem | logf1o 25075 | The natural logarithm function maps the nonzero complex numbers one-to-one onto its range. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | ||
Theorem | dfrelog 25076 | The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ (log ↾ ℝ+) = ◡(exp ↾ ℝ) | ||
Theorem | relogf1o 25077 | The natural logarithm function maps the positive reals one-to-one onto the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ (log ↾ ℝ+):ℝ+–1-1-onto→ℝ | ||
Theorem | logrncl 25078 | Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ran log) | ||
Theorem | logcl 25079 | Closure of the natural logarithm function. (Contributed by NM, 21-Apr-2008.) (Revised by Mario Carneiro, 23-Sep-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ) | ||
Theorem | logimcl 25080 | Closure of the imaginary part of the logarithm function. (Contributed by Mario Carneiro, 23-Sep-2014.) (Revised by Mario Carneiro, 1-Apr-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)) | ||
Theorem | logcld 25081 | The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 25079. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ≠ 0) ⇒ ⊢ (𝜑 → (log‘𝑋) ∈ ℂ) | ||
Theorem | logimcld 25082 | The imaginary part of the logarithm is in (-π(,]π). Deduction form of logimcl 25080. Compare logimclad 25083. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ≠ 0) ⇒ ⊢ (𝜑 → (-π < (ℑ‘(log‘𝑋)) ∧ (ℑ‘(log‘𝑋)) ≤ π)) | ||
Theorem | logimclad 25083 | The imaginary part of the logarithm is in (-π(,]π). Alternate form of logimcld 25082. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ≠ 0) ⇒ ⊢ (𝜑 → (ℑ‘(log‘𝑋)) ∈ (-π(,]π)) | ||
Theorem | abslogimle 25084 | The imaginary part of the logarithm function has absolute value less than pi. (Contributed by Mario Carneiro, 3-Jul-2017.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ≤ π) | ||
Theorem | logrnaddcl 25085 | The range of the natural logarithm is closed under addition with reals. (Contributed by Mario Carneiro, 3-Apr-2015.) |
⊢ ((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ran log) | ||
Theorem | relogcl 25086 | Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | ||
Theorem | eflog 25087 | Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴) | ||
Theorem | logeq0im1 25088 | If the logarithm of a number is 0, the number must be 1. (Contributed by David A. Wheeler, 22-Jul-2017.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (log‘𝐴) = 0) → 𝐴 = 1) | ||
Theorem | logccne0 25089 | The logarithm isn't 0 if its argument isn't 0 or 1. (Contributed by David A. Wheeler, 17-Jul-2017.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ≠ 0) | ||
Theorem | logne0 25090 | Logarithm of a non-1 positive real number is not zero and thus suitable as a divisor. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 14-Jun-2020.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≠ 1) → (log‘𝐴) ≠ 0) | ||
Theorem | reeflog 25091 | Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
⊢ (𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴) | ||
Theorem | logef 25092 | Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ (𝐴 ∈ ran log → (log‘(exp‘𝐴)) = 𝐴) | ||
Theorem | relogef 25093 | Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
⊢ (𝐴 ∈ ℝ → (log‘(exp‘𝐴)) = 𝐴) | ||
Theorem | logeftb 25094 | Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ran log) → ((log‘𝐴) = 𝐵 ↔ (exp‘𝐵) = 𝐴)) | ||
Theorem | relogeftb 25095 | Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ) → ((log‘𝐴) = 𝐵 ↔ (exp‘𝐵) = 𝐴)) | ||
Theorem | log1 25096 | The natural logarithm of 1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
⊢ (log‘1) = 0 | ||
Theorem | loge 25097 | The natural logarithm of e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
⊢ (log‘e) = 1 | ||
Theorem | logneg 25098 | The natural logarithm of a negative real number. (Contributed by Mario Carneiro, 13-May-2014.) (Revised by Mario Carneiro, 3-Apr-2015.) |
⊢ (𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π))) | ||
Theorem | logm1 25099 | The natural logarithm of negative 1. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.) |
⊢ (log‘-1) = (i · π) | ||
Theorem | lognegb 25100 | If a number has imaginary part equal to π, then it is on the negative real axis and vice-versa. (Contributed by Mario Carneiro, 23-Sep-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ+ ↔ (ℑ‘(log‘𝐴)) = π)) |
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