state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case inr
z : ℂ
h₀ : z ≠ 0
⊢ 0 ≤ arg z ↔ 0 ≤ z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | calc
0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) :=
⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by
contrapose!
intro h
exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩
_ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
rcases eq_or_ne z 0 with (rfl | h₀); · simp
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.166_0.CflASCTDE9UCom5 | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
h₀ : z ≠ 0
⊢ 0 ≤ Real.sin (arg z) → 0 ≤ arg z | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | contrapose! | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
rcases eq_or_ne z 0 with (rfl | h₀); · simp
calc
0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) :=
⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.166_0.CflASCTDE9UCom5 | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
h₀ : z ≠ 0
⊢ arg z < 0 → Real.sin (arg z) < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | intro h | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
rcases eq_or_ne z 0 with (rfl | h₀); · simp
calc
0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) :=
⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by
contrapose!
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.166_0.CflASCTDE9UCom5 | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
h₀ : z ≠ 0
h : arg z < 0
⊢ Real.sin (arg z) < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _) | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
rcases eq_or_ne z 0 with (rfl | h₀); · simp
calc
0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) :=
⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by
contrapose!
intro h
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.166_0.CflASCTDE9UCom5 | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
h₀ : z ≠ 0
⊢ 0 ≤ Real.sin (arg z) ↔ 0 ≤ z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
rcases eq_or_ne z 0 with (rfl | h₀); · simp
calc
0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) :=
⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by
contrapose!
intro h
exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.166_0.CflASCTDE9UCom5 | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
r : ℝ
hr : 0 < r
⊢ arg (↑r * x) = arg x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases eq_or_ne x 0 with (rfl | hx) | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.184_0.CflASCTDE9UCom5 | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
r : ℝ
hr : 0 < r
⊢ arg (↑r * 0) = arg 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [mul_zero] | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by
rcases eq_or_ne x 0 with (rfl | hx); · | Mathlib.Analysis.SpecialFunctions.Complex.Arg.184_0.CflASCTDE9UCom5 | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
x : ℂ
r : ℝ
hr : 0 < r
hx : x ≠ 0
⊢ arg (↑r * x) = arg x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | conv_lhs =>
rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul,
arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by
rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.184_0.CflASCTDE9UCom5 | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
r : ℝ
hr : 0 < r
hx : x ≠ 0
| arg (↑r * x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul,
arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by
rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero]
conv_lhs =>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.184_0.CflASCTDE9UCom5 | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
r : ℝ
hr : 0 < r
hx : x ≠ 0
| arg (↑r * x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul,
arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by
rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero]
conv_lhs =>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.184_0.CflASCTDE9UCom5 | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
r : ℝ
hr : 0 < r
hx : x ≠ 0
| arg (↑r * x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul,
arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by
rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero]
conv_lhs =>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.184_0.CflASCTDE9UCom5 | theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x y : ℂ
hx : x ≠ 0
hy : y ≠ 0
⊢ arg x = arg y ↔ ↑(abs y) / ↑(abs x) * x = y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs,
div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] | theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.194_0.CflASCTDE9UCom5 | theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
arg x = arg y ↔ (abs y / abs x : ℂ) * x = y | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x y : ℂ
hx : x ≠ 0
hy : y ≠ 0
⊢ arg x = arg y ↔ arg (↑(abs y) / ↑(abs x) * x) = arg y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← ofReal_div, arg_real_mul] | theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by
simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs,
div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.194_0.CflASCTDE9UCom5 | theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
arg x = arg y ↔ (abs y / abs x : ℂ) * x = y | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case hr
x y : ℂ
hx : x ≠ 0
hy : y ≠ 0
⊢ 0 < abs y / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact div_pos (abs.pos hy) (abs.pos hx) | theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by
simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs,
div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff]
rw [← ofReal_div, arg_real_mul]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.194_0.CflASCTDE9UCom5 | theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
arg x = arg y ↔ (abs y / abs x : ℂ) * x = y | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
⊢ arg 1 = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [arg, zero_le_one] | @[simp]
theorem arg_one : arg 1 = 0 := by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.202_0.CflASCTDE9UCom5 | @[simp]
theorem arg_one : arg 1 = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
⊢ arg (-1) = π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] | @[simp]
theorem arg_neg_one : arg (-1) = π := by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.206_0.CflASCTDE9UCom5 | @[simp]
theorem arg_neg_one : arg (-1) = π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
⊢ arg I = π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [arg, le_refl] | @[simp]
theorem arg_I : arg I = π / 2 := by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.210_0.CflASCTDE9UCom5 | @[simp]
theorem arg_I : arg I = π / 2 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
⊢ arg (-I) = -(π / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [arg, le_refl] | @[simp]
theorem arg_neg_I : arg (-I) = -(π / 2) := by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.215_0.CflASCTDE9UCom5 | @[simp]
theorem arg_neg_I : arg (-I) = -(π / 2) | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ Real.tan (arg x) = x.im / x.re | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | by_cases h : x = 0 | @[simp]
theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.220_0.CflASCTDE9UCom5 | @[simp]
theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
h : x = 0
⊢ Real.tan (arg x) = x.im / x.re | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] | @[simp]
theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by
by_cases h : x = 0
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.220_0.CflASCTDE9UCom5 | @[simp]
theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x : ℂ
h : ¬x = 0
⊢ Real.tan (arg x) = x.im / x.re | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] | @[simp]
theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by
by_cases h : x = 0
· simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.220_0.CflASCTDE9UCom5 | @[simp]
theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℝ
hx : 0 ≤ x
⊢ arg ↑x = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [arg, hx] | theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.227_0.CflASCTDE9UCom5 | theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | refine' ⟨fun h => _, _⟩ | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.230_0.CflASCTDE9UCom5 | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_1
z : ℂ
h : arg z = 0
⊢ 0 ≤ z.re ∧ z.im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← abs_mul_cos_add_sin_mul_I z, h] | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by
refine' ⟨fun h => _, _⟩
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.230_0.CflASCTDE9UCom5 | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_1
z : ℂ
h : arg z = 0
⊢ 0 ≤ (↑(abs z) * (cos ↑0 + sin ↑0 * I)).re ∧ (↑(abs z) * (cos ↑0 + sin ↑0 * I)).im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [abs.nonneg] | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by
refine' ⟨fun h => _, _⟩
· rw [← abs_mul_cos_add_sin_mul_I z, h]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.230_0.CflASCTDE9UCom5 | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2
z : ℂ
⊢ 0 ≤ z.re ∧ z.im = 0 → arg z = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | cases' z with x y | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by
refine' ⟨fun h => _, _⟩
· rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [abs.nonneg]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.230_0.CflASCTDE9UCom5 | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2.mk
x y : ℝ
⊢ 0 ≤ { re := x, im := y }.re ∧ { re := x, im := y }.im = 0 → arg { re := x, im := y } = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rintro ⟨h, rfl : y = 0⟩ | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by
refine' ⟨fun h => _, _⟩
· rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [abs.nonneg]
· cases' z with x y
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.230_0.CflASCTDE9UCom5 | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2.mk.intro
x : ℝ
h : 0 ≤ { re := x, im := 0 }.re
⊢ arg { re := x, im := 0 } = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact arg_ofReal_of_nonneg h | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by
refine' ⟨fun h => _, _⟩
· rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [abs.nonneg]
· cases' z with x y
rintro ⟨h, rfl : y = 0⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.230_0.CflASCTDE9UCom5 | theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ arg z = π ↔ z.re < 0 ∧ z.im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | by_cases h₀ : z = 0 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
z : ℂ
h₀ : z = 0
⊢ arg z = π ↔ z.re < 0 ∧ z.im = 0
case neg z : ℂ h₀ : ¬z = 0 ⊢ arg z = π ↔ z.re < 0 ∧ z.im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
by_cases h₀ : z = 0; | Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
z : ℂ
h₀ : ¬z = 0
⊢ arg z = π ↔ z.re < 0 ∧ z.im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | constructor | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mp
z : ℂ
h₀ : ¬z = 0
⊢ arg z = π → z.re < 0 ∧ z.im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | intro h | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm]
constructor
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mp
z : ℂ
h₀ : ¬z = 0
h : arg z = π
⊢ z.re < 0 ∧ z.im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← abs_mul_cos_add_sin_mul_I z, h] | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm]
constructor
· intro h
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mp
z : ℂ
h₀ : ¬z = 0
h : arg z = π
⊢ (↑(abs z) * (cos ↑π + sin ↑π * I)).re < 0 ∧ (↑(abs z) * (cos ↑π + sin ↑π * I)).im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [h₀] | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr
z : ℂ
h₀ : ¬z = 0
⊢ z.re < 0 ∧ z.im = 0 → arg z = π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | cases' z with x y | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr.mk
x y : ℝ
h₀ : ¬{ re := x, im := y } = 0
⊢ { re := x, im := y }.re < 0 ∧ { re := x, im := y }.im = 0 → arg { re := x, im := y } = π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rintro ⟨h : x < 0, rfl : y = 0⟩ | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· cases' z with x y
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr.mk.intro
x : ℝ
h : x < 0
h₀ : ¬{ re := x, im := 0 } = 0
⊢ arg { re := x, im := 0 } = π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· cases' z with x y
rintro ⟨h : x < 0, rfl : y = 0⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr.mk.intro
x : ℝ
h : x < 0
h₀ : ¬{ re := x, im := 0 } = 0
⊢ arg { re := x, im := 0 } = arg (↑(-x) * -1) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [← ofReal_def] | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by
by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· cases' z with x y
rintro ⟨h : x < 0, rfl : y = 0⟩
rw [← arg_neg_one, ← arg_real_mul (-1) (... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.239_0.CflASCTDE9UCom5 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] | theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.251_0.CflASCTDE9UCom5 | theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | by_cases h₀ : z = 0 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.259_0.CflASCTDE9UCom5 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
z : ℂ
h₀ : z = 0
⊢ arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by
by_cases h₀ : z = 0; · | Mathlib.Analysis.SpecialFunctions.Complex.Arg.259_0.CflASCTDE9UCom5 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
z : ℂ
h₀ : ¬z = 0
⊢ arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | constructor | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.259_0.CflASCTDE9UCom5 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mp
z : ℂ
h₀ : ¬z = 0
⊢ arg z = π / 2 → z.re = 0 ∧ 0 < z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | intro h | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne]
constructor
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.259_0.CflASCTDE9UCom5 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mp
z : ℂ
h₀ : ¬z = 0
h : arg z = π / 2
⊢ z.re = 0 ∧ 0 < z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← abs_mul_cos_add_sin_mul_I z, h] | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne]
constructor
· intro h
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.259_0.CflASCTDE9UCom5 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mp
z : ℂ
h₀ : ¬z = 0
h : arg z = π / 2
⊢ (↑(abs z) * (cos ↑(π / 2) + sin ↑(π / 2) * I)).re = 0 ∧ 0 < (↑(abs z) * (cos ↑(π / 2) + sin ↑(π / 2) * I)).im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [h₀] | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.259_0.CflASCTDE9UCom5 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr
z : ℂ
h₀ : ¬z = 0
⊢ z.re = 0 ∧ 0 < z.im → arg z = π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | cases' z with x y | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.259_0.CflASCTDE9UCom5 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr.mk
x y : ℝ
h₀ : ¬{ re := x, im := y } = 0
⊢ { re := x, im := y }.re = 0 ∧ 0 < { re := x, im := y }.im → arg { re := x, im := y } = π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rintro ⟨rfl : x = 0, hy : 0 < y⟩ | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· cases' z with x y
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.259_0.CflASCTDE9UCom5 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr.mk.intro
y : ℝ
h₀ : ¬{ re := 0, im := y } = 0
hy : 0 < y
⊢ arg { re := 0, im := y } = π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· cases' z with x y
rintro ⟨rfl : x = 0, hy : 0 < y⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.259_0.CflASCTDE9UCom5 | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | by_cases h₀ : z = 0 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
z : ℂ
h₀ : z = 0
⊢ arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [h₀, lt_irrefl, Real.pi_ne_zero] | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
by_cases h₀ : z = 0; · | Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
z : ℂ
h₀ : ¬z = 0
⊢ arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | constructor | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mp
z : ℂ
h₀ : ¬z = 0
⊢ arg z = -(π / 2) → z.re = 0 ∧ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | intro h | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero]
constructor
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mp
z : ℂ
h₀ : ¬z = 0
h : arg z = -(π / 2)
⊢ z.re = 0 ∧ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← abs_mul_cos_add_sin_mul_I z, h] | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero]
constructor
· intro h
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mp
z : ℂ
h₀ : ¬z = 0
h : arg z = -(π / 2)
⊢ (↑(abs z) * (cos ↑(-(π / 2)) + sin ↑(-(π / 2)) * I)).re = 0 ∧
(↑(abs z) * (cos ↑(-(π / 2)) + sin ↑(-(π / 2)) * I)).im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [h₀] | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr
z : ℂ
h₀ : ¬z = 0
⊢ z.re = 0 ∧ z.im < 0 → arg z = -(π / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | cases' z with x y | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr.mk
x y : ℝ
h₀ : ¬{ re := x, im := y } = 0
⊢ { re := x, im := y }.re = 0 ∧ { re := x, im := y }.im < 0 → arg { re := x, im := y } = -(π / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rintro ⟨rfl : x = 0, hy : y < 0⟩ | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· cases' z with x y
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr.mk.intro
y : ℝ
h₀ : ¬{ re := 0, im := y } = 0
hy : y < 0
⊢ arg { re := 0, im := y } = -(π / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· cases' z with x y
rintro ⟨rfl : x = 0, hy : y < 0⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.mpr.mk.intro
y : ℝ
h₀ : ¬{ re := 0, im := y } = 0
hy : y < 0
⊢ arg (↑0 + ↑y * I) = arg (↑(-y) * -I) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by
by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· cases' z with x y
rintro ⟨rfl : x = 0, hy : y < 0⟩
rw [← arg_neg_I, ← arg_... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.270_0.CflASCTDE9UCom5 | theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
hx_re : x.re < 0
hx_im : 0 ≤ x.im
⊢ arg x = arcsin ((-x).im / abs x) + π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [arg, hx_re.not_le, hx_im, if_true, if_false] | theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) :
arg x = Real.arcsin ((-x).im / abs x) + π := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.286_0.CflASCTDE9UCom5 | theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) :
arg x = Real.arcsin ((-x).im / abs x) + π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
hx_re : x.re < 0
hx_im : x.im < 0
⊢ arg x = arcsin ((-x).im / abs x) - π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [arg, hx_re.not_le, hx_im.not_le, if_false] | theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) :
arg x = Real.arcsin ((-x).im / abs x) - π := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.291_0.CflASCTDE9UCom5 | theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) :
arg x = Real.arcsin ((-x).im / abs x) - π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
h₁ : 0 ≤ z.im
h₂ : z ≠ 0
⊢ arg z = arccos (z.re / abs z) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] | theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) :
arg z = Real.arccos (z.re / abs z) := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.296_0.CflASCTDE9UCom5 | theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) :
arg z = Real.arccos (z.re / abs z) | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
hz : z.im < 0
⊢ arg z = -arccos (z.re / abs z) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne | theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.305_0.CflASCTDE9UCom5 | theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
hz : z.im < 0
h₀ : z ≠ 0
⊢ arg z = -arccos (z.re / abs z) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] | theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by
have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.305_0.CflASCTDE9UCom5 | theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case hx₁
z : ℂ
hz : z.im < 0
h₀ : z ≠ 0
⊢ 0 ≤ -arg z
case hx₂ z : ℂ hz : z.im < 0 h₀ : z ≠ 0 ⊢ -arg z ≤ π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] | theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by
have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne
rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.305_0.CflASCTDE9UCom5 | theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ arg ((starRingEnd ℂ) x) = if arg x = π then π else -arg x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / abs x) + π else -arcsin (x.im / abs x) - π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases lt_trichotomy x.re 0 with (hr | hr | hr) | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
x : ℂ
hr : x.re < 0
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / abs x) + π else -arcsin (x... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases lt_trichotomy x.im 0 with (hi | hi | hi) | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl
x : ℂ
hr : x.re = 0
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / abs x) + π else -arcsi... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases lt_trichotomy x.im 0 with (hi | hi | hi) | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inr
x : ℂ
hr : 0 < x.re
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / abs x) + π else -arcsi... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases lt_trichotomy x.im 0 with (hi | hi | hi) | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl.inl
x : ℂ
hr : x.re < 0
hi : x.im < 0
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / abs x) +... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl.inr.inl
x : ℂ
hr : x.re < 0
hi : x.im = 0
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / abs ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hr, hr.not_le, hi] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· simp [hr, hr.not_le, hi.... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl.inr.inr
x : ℂ
hr : x.re < 0
hi : 0 < x.im
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / abs ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· simp [hr, hr.not_le, hi.... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl.inl
x : ℂ
hr : x.re = 0
hi : x.im < 0
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / abs ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hr] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· simp [hr, hr.not_le, hi.... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl.inr.inl
x : ℂ
hr : x.re = 0
hi : x.im = 0
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hr] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· simp [hr, hr.not_le, hi.... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl.inr.inr
x : ℂ
hr : x.re = 0
hi : 0 < x.im
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hr] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· simp [hr, hr.not_le, hi.... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inr.inl
x : ℂ
hr : 0 < x.re
hi : x.im < 0
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / abs ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hr, hr.le, hi.ne] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· simp [hr, hr.not_le, hi.... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inr.inr.inl
x : ℂ
hr : 0 < x.re
hi : x.im = 0
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hr, hr.le, hr.le.not_lt] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· simp [hr, hr.not_le, hi.... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inr.inr.inr
x : ℂ
hr : 0 < x.re
hi : 0 < x.im
⊢ (if 0 ≤ x.re then -arcsin (x.im / abs x)
else if 0 ≤ -x.im then arcsin (x.im / abs x) + π else arcsin (x.im / abs x) - π) =
if x.re < 0 ∧ x.im = 0 then π
else
-if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then -arcsin (x.im / ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hr, hr.le, hr.le.not_lt] | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· simp [hr, hr.not_le, hi.... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.311_0.CflASCTDE9UCom5 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ arg x⁻¹ = if arg x = π then π else -arg x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← arg_conj, inv_def, mul_comm] | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.327_0.CflASCTDE9UCom5 | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ arg (↑(normSq x)⁻¹ * (starRingEnd ℂ) x) = arg ((starRingEnd ℂ) x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | by_cases hx : x = 0 | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by
rw [← arg_conj, inv_def, mul_comm]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.327_0.CflASCTDE9UCom5 | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
hx : x = 0
⊢ arg (↑(normSq x)⁻¹ * (starRingEnd ℂ) x) = arg ((starRingEnd ℂ) x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hx] | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by
rw [← arg_conj, inv_def, mul_comm]
by_cases hx : x = 0
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.327_0.CflASCTDE9UCom5 | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x : ℂ
hx : ¬x = 0
⊢ arg (↑(normSq x)⁻¹ * (starRingEnd ℂ) x) = arg ((starRingEnd ℂ) x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact arg_real_mul (conj x) (by simp [hx]) | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by
rw [← arg_conj, inv_def, mul_comm]
by_cases hx : x = 0
· simp [hx]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.327_0.CflASCTDE9UCom5 | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
hx : ¬x = 0
⊢ 0 < (normSq x)⁻¹ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hx] | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by
rw [← arg_conj, inv_def, mul_comm]
by_cases hx : x = 0
· simp [hx]
· exact arg_real_mul (conj x) (by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.327_0.CflASCTDE9UCom5 | theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ arg z ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases le_or_lt 0 (re z) with hre | hre | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
z : ℂ
hre : 0 ≤ z.re
⊢ arg z ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
rcases le_or_lt 0 (re z) with hre | hre
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
z : ℂ
hre : z.re < 0
⊢ arg z ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [hre.not_le, false_or_iff] | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
z : ℂ
hre : z.re < 0
⊢ arg z ≤ π / 2 ↔ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases le_or_lt 0 (im z) with him | him | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]
simp only [hre.not_le, false_or_iff]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl
z : ℂ
hre : z.re < 0
him : 0 ≤ z.im
⊢ arg z ≤ π / 2 ↔ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [him.not_lt] | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl
z : ℂ
hre : z.re < 0
him : 0 ≤ z.im
⊢ arg z ≤ π / 2 ↔ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub,
Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ←
_root_.abs_of_nonneg him, abs_im_lt_abs] | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· simp only [him.not_lt]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl
z : ℂ
hre : z.re < 0
him : 0 ≤ z.im
⊢ z.re ≠ 0
case inr.inl z : ℂ hre : z.re < 0 him : 0 ≤ z.im ⊢ 0 < abs z | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· simp only [him.not_lt]
rw... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inr
z : ℂ
hre : z.re < 0
him : z.im < 0
⊢ arg z ≤ π / 2 ↔ z.im < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [him] | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· simp only [him.not_lt]
rw... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inr
z : ℂ
hre : z.re < 0
him : z.im < 0
⊢ arg z ≤ π / 2 ↔ True | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· simp only [him.not_lt]
rw... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inr
z : ℂ
hre : z.re < 0
him : z.im < 0
⊢ arcsin ((-z).im / abs z) - π ≤ π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· simp only [him.not_lt]
rw... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.334_0.CflASCTDE9UCom5 | theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ -(π / 2) ≤ arg z ↔ 0 ≤ z.re ∨ 0 ≤ z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases le_or_lt 0 (re z) with hre | hre | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5 | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
z : ℂ
hre : 0 ≤ z.re
⊢ -(π / 2) ≤ arg z ↔ 0 ≤ z.re ∨ 0 ≤ z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by
rcases le_or_lt 0 (re z) with hre | hre
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5 | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
z : ℂ
hre : z.re < 0
⊢ -(π / 2) ≤ arg z ↔ 0 ≤ z.re ∨ 0 ≤ z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [hre.not_le, false_or_iff] | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5 | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
z : ℂ
hre : z.re < 0
⊢ -(π / 2) ≤ arg z ↔ 0 ≤ z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases le_or_lt 0 (im z) with him | him | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff]
simp only [hre.not_le, false_or_iff]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5 | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl
z : ℂ
hre : z.re < 0
him : 0 ≤ z.im
⊢ -(π / 2) ≤ arg z ↔ 0 ≤ z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [him] | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5 | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl
z : ℂ
hre : z.re < 0
him : 0 ≤ z.im
⊢ -(π / 2) ≤ arg z ↔ True | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· simp only [him]
... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5 | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.inl
z : ℂ
hre : z.re < 0
him : 0 ≤ z.im
⊢ -(π / 2) ≤ arcsin ((-z).im / abs z) + π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· simp only [him]
... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5 | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.