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 |
|---|---|---|---|---|---|---|
x y : ℂ
⊢ (cexp ↑x.re * (cos ↑x.im + sin ↑x.im * I)).im = Real.exp x.re * Real.sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [exp_ofReal_re, sin_ofReal_re] | theorem exp_im : (exp x).im = Real.exp x.re * Real.sin x.im := by
rw [exp_eq_exp_re_mul_sin_add_cos]
| Mathlib.Data.Complex.Exponential.1103_0.1bixbwhfBeJKySp | theorem exp_im : (exp x).im = Real.exp x.re * Real.sin x.im | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (cexp (↑x * I)).re = Real.cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [exp_mul_I, cos_ofReal_re] | @[simp]
theorem exp_ofReal_mul_I_re (x : ℝ) : (exp (x * I)).re = Real.cos x := by
| Mathlib.Data.Complex.Exponential.1108_0.1bixbwhfBeJKySp | @[simp]
theorem exp_ofReal_mul_I_re (x : ℝ) : (exp (x * I)).re = Real.cos x | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (cexp (↑x * I)).im = Real.sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [exp_mul_I, sin_ofReal_re] | @[simp]
theorem exp_ofReal_mul_I_im (x : ℝ) : (exp (x * I)).im = Real.sin x := by
| Mathlib.Data.Complex.Exponential.1114_0.1bixbwhfBeJKySp | @[simp]
theorem exp_ofReal_mul_I_im (x : ℝ) : (exp (x * I)).im = Real.sin x | Mathlib_Data_Complex_Exponential |
x y : ℂ
n : ℕ
z : ℂ
⊢ (cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← exp_mul_I, ← exp_mul_I] | /-- **De Moivre's formula** -/
theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
(cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I := by
| Mathlib.Data.Complex.Exponential.1120_0.1bixbwhfBeJKySp | /-- **De Moivre's formula** -/
theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
(cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
n : ℕ
z : ℂ
⊢ cexp (z * I) ^ n = cexp (↑n * z * 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | induction' n with n ih | /-- **De Moivre's formula** -/
theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
(cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I := by
rw [← exp_mul_I, ← exp_mul_I]
| Mathlib.Data.Complex.Exponential.1120_0.1bixbwhfBeJKySp | /-- **De Moivre's formula** -/
theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
(cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I | Mathlib_Data_Complex_Exponential |
case zero
x y z : ℂ
⊢ cexp (z * I) ^ Nat.zero = cexp (↑Nat.zero * z * 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [pow_zero, Nat.cast_zero, zero_mul, zero_mul, exp_zero] | /-- **De Moivre's formula** -/
theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
(cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I := by
rw [← exp_mul_I, ← exp_mul_I]
induction' n with n ih
· | Mathlib.Data.Complex.Exponential.1120_0.1bixbwhfBeJKySp | /-- **De Moivre's formula** -/
theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
(cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I | Mathlib_Data_Complex_Exponential |
case succ
x y z : ℂ
n : ℕ
ih : cexp (z * I) ^ n = cexp (↑n * z * I)
⊢ cexp (z * I) ^ Nat.succ n = cexp (↑(Nat.succ n) * z * 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [pow_succ', ih, Nat.cast_succ, add_mul, add_mul, one_mul, exp_add] | /-- **De Moivre's formula** -/
theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
(cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I := by
rw [← exp_mul_I, ← exp_mul_I]
induction' n with n ih
· rw [pow_zero, Nat.cast_zero, zero_mul, zero_mul, exp_zero]
· | Mathlib.Data.Complex.Exponential.1120_0.1bixbwhfBeJKySp | /-- **De Moivre's formula** -/
theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
(cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ rexp 0 = 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [Real.exp] | @[simp]
theorem exp_zero : exp 0 = 1 := by | Mathlib.Data.Complex.Exponential.1138_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ rexp (x + y) = rexp x * rexp 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [exp_add, exp] | nonrec theorem exp_add : exp (x + y) = exp x * exp y := by | Mathlib.Data.Complex.Exponential.1142_0.1bixbwhfBeJKySp | nonrec theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ (fun x => rexp (Multiplicative.toAdd x)) 1 = 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | /-- the exponential function as a monoid hom from `Multiplicative ℝ` to `ℝ` -/
noncomputable def expMonoidHom : MonoidHom (Multiplicative ℝ) ℝ :=
{ toFun := fun x => exp (Multiplicative.toAdd x),
map_one' := by | Mathlib.Data.Complex.Exponential.1146_0.1bixbwhfBeJKySp | /-- the exponential function as a monoid hom from `Multiplicative ℝ` to `ℝ` -/
noncomputable def expMonoidHom : MonoidHom (Multiplicative ℝ) ℝ | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ∀ (x y : Multiplicative ℝ),
OneHom.toFun { toFun := fun x => rexp (Multiplicative.toAdd x), map_one' := (_ : rexp 0 = 1) } (x * y) =
OneHom.toFun { toFun := fun x => rexp (Multiplicative.toAdd x), map_one' := (_ : rexp 0 = 1) } x *
OneHom.toFun { toFun := fun x => rexp (Multiplicative.toAdd ... | /-
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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [exp_add] | /-- the exponential function as a monoid hom from `Multiplicative ℝ` to `ℝ` -/
noncomputable def expMonoidHom : MonoidHom (Multiplicative ℝ) ℝ :=
{ toFun := fun x => exp (Multiplicative.toAdd x),
map_one' := by simp,
map_mul' := by | Mathlib.Data.Complex.Exponential.1146_0.1bixbwhfBeJKySp | /-- the exponential function as a monoid hom from `Multiplicative ℝ` to `ℝ` -/
noncomputable def expMonoidHom : MonoidHom (Multiplicative ℝ) ℝ | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
n : ℕ
⊢ ↑(rexp (↑n * x)) = ↑(rexp x ^ n) | /-
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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [exp_nat_mul] | nonrec theorem exp_nat_mul (x : ℝ) (n : ℕ) : exp (n * x) = exp x ^ n :=
ofReal_injective (by | Mathlib.Data.Complex.Exponential.1168_0.1bixbwhfBeJKySp | nonrec theorem exp_nat_mul (x : ℝ) (n : ℕ) : exp (n * x) = exp x ^ n | Mathlib_Data_Complex_Exponential |
x y : ℝ
h : rexp x = 0
⊢ cexp ↑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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [exp, ← ofReal_inj] at h | nonrec theorem exp_ne_zero : exp x ≠ 0 := fun h =>
exp_ne_zero x <| by | Mathlib.Data.Complex.Exponential.1172_0.1bixbwhfBeJKySp | nonrec theorem exp_ne_zero : exp x ≠ 0 | Mathlib_Data_Complex_Exponential |
x y : ℝ
h : ↑(cexp ↑x).re = ↑0
⊢ cexp ↑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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp_all | nonrec theorem exp_ne_zero : exp x ≠ 0 := fun h =>
exp_ne_zero x <| by rw [exp, ← ofReal_inj] at h; | Mathlib.Data.Complex.Exponential.1172_0.1bixbwhfBeJKySp | nonrec theorem exp_ne_zero : exp x ≠ 0 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(rexp (-x)) = ↑(rexp 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [exp_neg] | nonrec theorem exp_neg : exp (-x) = (exp x)⁻¹ :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1176_0.1bixbwhfBeJKySp | nonrec theorem exp_neg : exp (-x) = (exp x)⁻¹ | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ rexp (x - y) = rexp x / rexp 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv] | theorem exp_sub : exp (x - y) = exp x / exp y := by
| Mathlib.Data.Complex.Exponential.1180_0.1bixbwhfBeJKySp | theorem exp_sub : exp (x - y) = exp x / exp y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sin] | @[simp]
theorem sin_zero : sin 0 = 0 := by | Mathlib.Data.Complex.Exponential.1184_0.1bixbwhfBeJKySp | @[simp]
theorem sin_zero : sin 0 = 0 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sin (-x) = -sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sin, exp_neg, (neg_div _ _).symm, add_mul] | @[simp]
theorem sin_neg : sin (-x) = -sin x := by | Mathlib.Data.Complex.Exponential.1188_0.1bixbwhfBeJKySp | @[simp]
theorem sin_neg : sin (-x) = -sin x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(sin (x + y)) = ↑(sin x * cos y + cos x * sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sin_add] | nonrec theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1192_0.1bixbwhfBeJKySp | nonrec theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos 0 = 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos] | @[simp]
theorem cos_zero : cos 0 = 1 := by | Mathlib.Data.Complex.Exponential.1196_0.1bixbwhfBeJKySp | @[simp]
theorem cos_zero : cos 0 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos (-x) = cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos, exp_neg] | @[simp]
theorem cos_neg : cos (-x) = cos x := by | Mathlib.Data.Complex.Exponential.1200_0.1bixbwhfBeJKySp | @[simp]
theorem cos_neg : cos (-x) = cos x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos (abs' x) = cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | cases le_total x 0 | @[simp]
theorem cos_abs : cos |x| = cos x := by
| Mathlib.Data.Complex.Exponential.1204_0.1bixbwhfBeJKySp | @[simp]
theorem cos_abs : cos |x| = cos x | Mathlib_Data_Complex_Exponential |
case inl
x y : ℝ
h✝ : x ≤ 0
⊢ cos (abs' x) = cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [*, _root_.abs_of_nonneg, abs_of_nonpos, cos_neg] | @[simp]
theorem cos_abs : cos |x| = cos x := by
cases le_total x 0 <;> | Mathlib.Data.Complex.Exponential.1204_0.1bixbwhfBeJKySp | @[simp]
theorem cos_abs : cos |x| = cos x | Mathlib_Data_Complex_Exponential |
case inr
x y : ℝ
h✝ : 0 ≤ x
⊢ cos (abs' x) = cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [*, _root_.abs_of_nonneg, abs_of_nonpos, cos_neg] | @[simp]
theorem cos_abs : cos |x| = cos x := by
cases le_total x 0 <;> | Mathlib.Data.Complex.Exponential.1204_0.1bixbwhfBeJKySp | @[simp]
theorem cos_abs : cos |x| = cos x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cos (x + y)) = ↑(cos x * cos y - sin x * sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos_add] | nonrec theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1209_0.1bixbwhfBeJKySp | nonrec theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sin (x - y) = sin x * cos y - cos x * sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg] | theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y := by
| Mathlib.Data.Complex.Exponential.1213_0.1bixbwhfBeJKySp | theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos (x - y) = cos x * cos y + sin x * sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg] | theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y := by
| Mathlib.Data.Complex.Exponential.1217_0.1bixbwhfBeJKySp | theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(sin x - sin y) = ↑(2 * sin ((x - y) / 2) * cos ((x + y) / 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sin_sub_sin] | nonrec theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1221_0.1bixbwhfBeJKySp | nonrec theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cos x - cos y) = ↑(-2 * sin ((x + y) / 2) * sin ((x - y) / 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos_sub_cos] | nonrec theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1225_0.1bixbwhfBeJKySp | nonrec theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cos x + cos y) = ↑(2 * cos ((x + y) / 2) * cos ((x - y) / 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos_add_cos] | nonrec theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1229_0.1bixbwhfBeJKySp | nonrec theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(tan x) = ↑(sin x / cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [tan_eq_sin_div_cos] | nonrec theorem tan_eq_sin_div_cos : tan x = sin x / cos x :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1233_0.1bixbwhfBeJKySp | nonrec theorem tan_eq_sin_div_cos : tan x = sin x / cos x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : cos x ≠ 0
⊢ tan x * cos x = sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [tan_eq_sin_div_cos, div_mul_cancel _ hx] | theorem tan_mul_cos {x : ℝ} (hx : cos x ≠ 0) : tan x * cos x = sin x := by
| Mathlib.Data.Complex.Exponential.1237_0.1bixbwhfBeJKySp | theorem tan_mul_cos {x : ℝ} (hx : cos x ≠ 0) : tan x * cos x = sin x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ tan 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [tan] | @[simp]
theorem tan_zero : tan 0 = 0 := by | Mathlib.Data.Complex.Exponential.1241_0.1bixbwhfBeJKySp | @[simp]
theorem tan_zero : tan 0 = 0 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ tan (-x) = -tan 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [tan, neg_div] | @[simp]
theorem tan_neg : tan (-x) = -tan x := by | Mathlib.Data.Complex.Exponential.1245_0.1bixbwhfBeJKySp | @[simp]
theorem tan_neg : tan (-x) = -tan x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(sin x ^ 2 + cos x ^ 2) = ↑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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sin_sq_add_cos_sq] | @[simp]
nonrec theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=
ofReal_injective (by | Mathlib.Data.Complex.Exponential.1249_0.1bixbwhfBeJKySp | @[simp]
nonrec theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos x ^ 2 + sin x ^ 2 = 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [add_comm, sin_sq_add_cos_sq] | @[simp]
theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 := by | Mathlib.Data.Complex.Exponential.1254_0.1bixbwhfBeJKySp | @[simp]
theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sin x ^ 2 ≤ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← sin_sq_add_cos_sq x] | theorem sin_sq_le_one : sin x ^ 2 ≤ 1 := by
| Mathlib.Data.Complex.Exponential.1258_0.1bixbwhfBeJKySp | theorem sin_sq_le_one : sin x ^ 2 ≤ 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sin x ^ 2 ≤ sin x ^ 2 + cos x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact le_add_of_nonneg_right (sq_nonneg _) | theorem sin_sq_le_one : sin x ^ 2 ≤ 1 := by
rw [← sin_sq_add_cos_sq x]; | Mathlib.Data.Complex.Exponential.1258_0.1bixbwhfBeJKySp | theorem sin_sq_le_one : sin x ^ 2 ≤ 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos x ^ 2 ≤ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← sin_sq_add_cos_sq x] | theorem cos_sq_le_one : cos x ^ 2 ≤ 1 := by
| Mathlib.Data.Complex.Exponential.1262_0.1bixbwhfBeJKySp | theorem cos_sq_le_one : cos x ^ 2 ≤ 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos x ^ 2 ≤ sin x ^ 2 + cos x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact le_add_of_nonneg_left (sq_nonneg _) | theorem cos_sq_le_one : cos x ^ 2 ≤ 1 := by
rw [← sin_sq_add_cos_sq x]; | Mathlib.Data.Complex.Exponential.1262_0.1bixbwhfBeJKySp | theorem cos_sq_le_one : cos x ^ 2 ≤ 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sin x * sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [← sq, sin_sq_le_one] | theorem abs_sin_le_one : |sin x| ≤ 1 :=
abs_le_one_iff_mul_self_le_one.2 <| by | Mathlib.Data.Complex.Exponential.1266_0.1bixbwhfBeJKySp | theorem abs_sin_le_one : |sin x| ≤ 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos x * cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [← sq, cos_sq_le_one] | theorem abs_cos_le_one : |cos x| ≤ 1 :=
abs_le_one_iff_mul_self_le_one.2 <| by | Mathlib.Data.Complex.Exponential.1270_0.1bixbwhfBeJKySp | theorem abs_cos_le_one : |cos x| ≤ 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cos (2 * x)) = ↑(2 * cos x ^ 2 - 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos_two_mul] | nonrec theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1290_0.1bixbwhfBeJKySp | nonrec theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cos (2 * x)) = ↑(cos x ^ 2 - sin x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos_two_mul'] | nonrec theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1294_0.1bixbwhfBeJKySp | nonrec theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(sin (2 * x)) = ↑(2 * sin x * cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sin_two_mul] | nonrec theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1298_0.1bixbwhfBeJKySp | nonrec theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cos x ^ 2) = ↑(1 / 2 + cos (2 * x) / 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos_sq] | nonrec theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1302_0.1bixbwhfBeJKySp | nonrec theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos x ^ 2 = 1 - sin x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← sin_sq_add_cos_sq x, add_sub_cancel'] | theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 := by | Mathlib.Data.Complex.Exponential.1306_0.1bixbwhfBeJKySp | theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sin x ^ 2 = 1 / 2 - cos (2 * x) / 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [sin_sq, cos_sq, ← sub_sub, sub_half] | lemma sin_sq_eq_half_sub : sin x ^ 2 = 1 / 2 - cos (2 * x) / 2 := by
| Mathlib.Data.Complex.Exponential.1313_0.1bixbwhfBeJKySp | lemma sin_sq_eq_half_sub : sin x ^ 2 = 1 / 2 - cos (2 * x) / 2 | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
⊢ abs' (sin x) = sqrt (1 - cos x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← sin_sq, sqrt_sq_eq_abs] | theorem abs_sin_eq_sqrt_one_sub_cos_sq (x : ℝ) : |sin x| = sqrt (1 - cos x ^ 2) := by
| Mathlib.Data.Complex.Exponential.1316_0.1bixbwhfBeJKySp | theorem abs_sin_eq_sqrt_one_sub_cos_sq (x : ℝ) : |sin x| = sqrt (1 - cos x ^ 2) | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
⊢ abs' (cos x) = sqrt (1 - sin x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← cos_sq', sqrt_sq_eq_abs] | theorem abs_cos_eq_sqrt_one_sub_sin_sq (x : ℝ) : |cos x| = sqrt (1 - sin x ^ 2) := by
| Mathlib.Data.Complex.Exponential.1320_0.1bixbwhfBeJKySp | theorem abs_cos_eq_sqrt_one_sub_sin_sq (x : ℝ) : |cos x| = sqrt (1 - sin x ^ 2) | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : cos x ≠ 0
this : Complex.cos ↑x ≠ 0
⊢ ↑(1 + tan x ^ 2)⁻¹ = ↑(cos x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa using Complex.inv_one_add_tan_sq this | theorem inv_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 :=
have : Complex.cos x ≠ 0 := mt (congr_arg re) hx
ofReal_inj.1 <| by | Mathlib.Data.Complex.Exponential.1324_0.1bixbwhfBeJKySp | theorem inv_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : cos x ≠ 0
⊢ tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul] | theorem tan_sq_div_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) :
tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 := by
| Mathlib.Data.Complex.Exponential.1329_0.1bixbwhfBeJKySp | theorem tan_sq_div_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) :
tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 < cos x
⊢ (sqrt (1 + tan x ^ 2))⁻¹ = cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← sqrt_sq hx.le, ← sqrt_inv, inv_one_add_tan_sq hx.ne'] | theorem inv_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) : (sqrt (1 + tan x ^ 2))⁻¹ = cos x := by
| Mathlib.Data.Complex.Exponential.1334_0.1bixbwhfBeJKySp | theorem inv_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) : (sqrt (1 + tan x ^ 2))⁻¹ = cos x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 < cos x
⊢ tan x / sqrt (1 + tan x ^ 2) = sin 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← tan_mul_cos hx.ne', ← inv_sqrt_one_add_tan_sq hx, div_eq_mul_inv] | theorem tan_div_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) :
tan x / sqrt (1 + tan x ^ 2) = sin x := by
| Mathlib.Data.Complex.Exponential.1338_0.1bixbwhfBeJKySp | theorem tan_div_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) :
tan x / sqrt (1 + tan x ^ 2) = sin x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cos (3 * x) = 4 * cos x ^ 3 - 3 * cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
| Mathlib.Data.Complex.Exponential.1343_0.1bixbwhfBeJKySp | nonrec theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cos (3 * x)) = ↑(4 * cos x ^ 3 - 3 * cos 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos_three_mul] | nonrec theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1343_0.1bixbwhfBeJKySp | nonrec theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | /-
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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
| Mathlib.Data.Complex.Exponential.1347_0.1bixbwhfBeJKySp | nonrec theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(sin (3 * x)) = ↑(3 * sin x - 4 * sin x ^ 3) | /-
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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sin_three_mul] | nonrec theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1347_0.1bixbwhfBeJKySp | nonrec theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
⊢ ↑(sinh x) = ↑((rexp x - rexp (-x)) / 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [Complex.sinh] | /-- The definition of `sinh` in terms of `exp`. -/
nonrec theorem sinh_eq (x : ℝ) : sinh x = (exp x - exp (-x)) / 2 :=
ofReal_injective <| by | Mathlib.Data.Complex.Exponential.1351_0.1bixbwhfBeJKySp | /-- The definition of `sinh` in terms of `exp`. -/
nonrec theorem sinh_eq (x : ℝ) : sinh x = (exp x - exp (-x)) / 2 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sinh] | @[simp]
theorem sinh_zero : sinh 0 = 0 := by | Mathlib.Data.Complex.Exponential.1356_0.1bixbwhfBeJKySp | @[simp]
theorem sinh_zero : sinh 0 = 0 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sinh (-x) = -sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sinh, exp_neg, (neg_div _ _).symm, add_mul] | @[simp]
theorem sinh_neg : sinh (-x) = -sinh x := by | Mathlib.Data.Complex.Exponential.1360_0.1bixbwhfBeJKySp | @[simp]
theorem sinh_neg : sinh (-x) = -sinh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sinh (x + y) = sinh x * cosh y + cosh x * sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y := by
| Mathlib.Data.Complex.Exponential.1364_0.1bixbwhfBeJKySp | nonrec theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(sinh (x + y)) = ↑(sinh x * cosh y + cosh x * sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sinh_add] | nonrec theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y := by
rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1364_0.1bixbwhfBeJKySp | nonrec theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
⊢ cosh x * 2 = rexp x + rexp (-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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [cosh, exp, exp, Complex.ofReal_neg, Complex.cosh, mul_two, ← Complex.add_re, ← mul_two,
div_mul_cancel _ (two_ne_zero' ℂ), Complex.add_re] | /-- The definition of `cosh` in terms of `exp`. -/
theorem cosh_eq (x : ℝ) : cosh x = (exp x + exp (-x)) / 2 :=
eq_div_of_mul_eq two_ne_zero <| by
| Mathlib.Data.Complex.Exponential.1368_0.1bixbwhfBeJKySp | /-- The definition of `cosh` in terms of `exp`. -/
theorem cosh_eq (x : ℝ) : cosh x = (exp x + exp (-x)) / 2 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cosh 0 = 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cosh] | @[simp]
theorem cosh_zero : cosh 0 = 1 := by | Mathlib.Data.Complex.Exponential.1375_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_zero : cosh 0 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cosh (-x)) = ↑(cosh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | @[simp]
theorem cosh_neg : cosh (-x) = cosh x :=
ofReal_inj.1 <| by | Mathlib.Data.Complex.Exponential.1379_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_neg : cosh (-x) = cosh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cosh (abs' x) = cosh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | cases le_total x 0 | @[simp]
theorem cosh_abs : cosh |x| = cosh x := by
| Mathlib.Data.Complex.Exponential.1384_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_abs : cosh |x| = cosh x | Mathlib_Data_Complex_Exponential |
case inl
x y : ℝ
h✝ : x ≤ 0
⊢ cosh (abs' x) = cosh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [*, _root_.abs_of_nonneg, abs_of_nonpos] | @[simp]
theorem cosh_abs : cosh |x| = cosh x := by
cases le_total x 0 <;> | Mathlib.Data.Complex.Exponential.1384_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_abs : cosh |x| = cosh x | Mathlib_Data_Complex_Exponential |
case inr
x y : ℝ
h✝ : 0 ≤ x
⊢ cosh (abs' x) = cosh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [*, _root_.abs_of_nonneg, abs_of_nonpos] | @[simp]
theorem cosh_abs : cosh |x| = cosh x := by
cases le_total x 0 <;> | Mathlib.Data.Complex.Exponential.1384_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_abs : cosh |x| = cosh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cosh (x + y) = cosh x * cosh y + sinh x * sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y := by
| Mathlib.Data.Complex.Exponential.1389_0.1bixbwhfBeJKySp | nonrec theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cosh (x + y)) = ↑(cosh x * cosh y + sinh x * sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cosh_add] | nonrec theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y := by
rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1389_0.1bixbwhfBeJKySp | nonrec theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sinh (x - y) = sinh x * cosh y - cosh x * sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg] | theorem sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y := by
| Mathlib.Data.Complex.Exponential.1393_0.1bixbwhfBeJKySp | theorem sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cosh (x - y) = cosh x * cosh y - sinh x * sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg] | theorem cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y := by
| Mathlib.Data.Complex.Exponential.1397_0.1bixbwhfBeJKySp | theorem cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(tanh x) = ↑(sinh x / cosh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [tanh_eq_sinh_div_cosh] | nonrec theorem tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x :=
ofReal_inj.1 <| by | Mathlib.Data.Complex.Exponential.1401_0.1bixbwhfBeJKySp | nonrec theorem tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ tanh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [tanh] | @[simp]
theorem tanh_zero : tanh 0 = 0 := by | Mathlib.Data.Complex.Exponential.1405_0.1bixbwhfBeJKySp | @[simp]
theorem tanh_zero : tanh 0 = 0 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ tanh (-x) = -tanh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [tanh, neg_div] | @[simp]
theorem tanh_neg : tanh (-x) = -tanh x := by | Mathlib.Data.Complex.Exponential.1409_0.1bixbwhfBeJKySp | @[simp]
theorem tanh_neg : tanh (-x) = -tanh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cosh x + sinh x = rexp 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | @[simp]
theorem cosh_add_sinh : cosh x + sinh x = exp x := by | Mathlib.Data.Complex.Exponential.1413_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_add_sinh : cosh x + sinh x = exp x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cosh x + sinh x) = ↑(rexp 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | @[simp]
theorem cosh_add_sinh : cosh x + sinh x = exp x := by rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1413_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_add_sinh : cosh x + sinh x = exp x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sinh x + cosh x = rexp 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [add_comm, cosh_add_sinh] | @[simp]
theorem sinh_add_cosh : sinh x + cosh x = exp x := by | Mathlib.Data.Complex.Exponential.1417_0.1bixbwhfBeJKySp | @[simp]
theorem sinh_add_cosh : sinh x + cosh x = exp x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cosh x - sinh x = rexp (-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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | @[simp]
theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) := by
| Mathlib.Data.Complex.Exponential.1431_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cosh x - sinh x) = ↑(rexp (-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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | @[simp]
theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) := by
rw [← ofReal_inj]
| Mathlib.Data.Complex.Exponential.1431_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sinh x - cosh x = -rexp (-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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← neg_sub, cosh_sub_sinh] | @[simp]
theorem sinh_sub_cosh : sinh x - cosh x = -exp (-x) := by | Mathlib.Data.Complex.Exponential.1437_0.1bixbwhfBeJKySp | @[simp]
theorem sinh_sub_cosh : sinh x - cosh x = -exp (-x) | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
⊢ cosh x ^ 2 - sinh x ^ 2 = 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | @[simp]
theorem cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 := by | Mathlib.Data.Complex.Exponential.1441_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
⊢ ↑(cosh x ^ 2 - sinh x ^ 2) = ↑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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | @[simp]
theorem cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 := by rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1441_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cosh x ^ 2 = sinh x ^ 2 + 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 := by | Mathlib.Data.Complex.Exponential.1445_0.1bixbwhfBeJKySp | nonrec theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cosh x ^ 2) = ↑(sinh x ^ 2 + 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cosh_sq] | nonrec theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 := by rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1445_0.1bixbwhfBeJKySp | nonrec theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sinh x ^ 2 = cosh x ^ 2 - 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by | Mathlib.Data.Complex.Exponential.1452_0.1bixbwhfBeJKySp | nonrec theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(sinh x ^ 2) = ↑(cosh x ^ 2 - 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sinh_sq] | nonrec theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1452_0.1bixbwhfBeJKySp | nonrec theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cosh (2 * x) = cosh x ^ 2 + sinh x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 := by
| Mathlib.Data.Complex.Exponential.1455_0.1bixbwhfBeJKySp | nonrec theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cosh (2 * x)) = ↑(cosh x ^ 2 + sinh x ^ 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cosh_two_mul] | nonrec theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 := by
rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1455_0.1bixbwhfBeJKySp | nonrec theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sinh (2 * x) = 2 * sinh x * cosh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x := by
| Mathlib.Data.Complex.Exponential.1459_0.1bixbwhfBeJKySp | nonrec theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(sinh (2 * x)) = ↑(2 * sinh x * cosh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sinh_two_mul] | nonrec theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x := by
rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1459_0.1bixbwhfBeJKySp | nonrec theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
| Mathlib.Data.Complex.Exponential.1463_0.1bixbwhfBeJKySp | nonrec theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(cosh (3 * x)) = ↑(4 * cosh x ^ 3 - 3 * cosh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cosh_three_mul] | nonrec theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1463_0.1bixbwhfBeJKySp | nonrec theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_inj] | nonrec theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
| Mathlib.Data.Complex.Exponential.1467_0.1bixbwhfBeJKySp | nonrec theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
x y : ℝ
⊢ ↑(sinh (3 * x)) = ↑(4 * sinh x ^ 3 + 3 * sinh 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sinh_three_mul] | nonrec theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
rw [← ofReal_inj]; | Mathlib.Data.Complex.Exponential.1467_0.1bixbwhfBeJKySp | nonrec theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 ≤ x
n : ℕ
⊢ ∑ i in range n, x ^ i / ↑i ! ≤
CauSeq.lim { val := fun n => (↑(exp' ↑x) n).re, property := (_ : IsCauSeq abs' fun n => (↑(exp' ↑x) n).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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | refine' le_lim (CauSeq.le_of_exists ⟨n, fun j hj => _⟩) | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x :=
calc
∑ i in range n, x ^ i / i ! ≤ lim (⟨_, isCauSeq_re (exp' x)⟩ : CauSeq ℝ Abs.abs) := by
| Mathlib.Data.Complex.Exponential.1473_0.1bixbwhfBeJKySp | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 ≤ x
n j : ℕ
hj : j ≥ n
⊢ ↑(const abs' (∑ i in range n, x ^ i / ↑i !)) j ≤
↑{ val := fun n => (↑(exp' ↑x) n).re, property := (_ : IsCauSeq abs' fun n => (↑(exp' ↑x) n).re) } j | /-
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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [exp', const_apply, re_sum] | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x :=
calc
∑ i in range n, x ^ i / i ! ≤ lim (⟨_, isCauSeq_re (exp' x)⟩ : CauSeq ℝ Abs.abs) := by
refine' le_lim (CauSeq.le_of_exists ⟨n, fun j hj => _⟩)
| Mathlib.Data.Complex.Exponential.1473_0.1bixbwhfBeJKySp | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 ≤ x
n j : ℕ
hj : j ≥ n
⊢ ∑ i in range n, x ^ i / ↑i ! ≤ ∑ i in range j, (↑x ^ i / ↑i !).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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_cast | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x :=
calc
∑ i in range n, x ^ i / i ! ≤ lim (⟨_, isCauSeq_re (exp' x)⟩ : CauSeq ℝ Abs.abs) := by
refine' le_lim (CauSeq.le_of_exists ⟨n, fun j hj => _⟩)
simp only [exp', const_apply, re_sum]
| Mathlib.Data.Complex.Exponential.1473_0.1bixbwhfBeJKySp | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 ≤ x
n j : ℕ
hj : j ≥ n
⊢ ∑ i in range n, x ^ i / ↑i ! ≤ ∑ i in range j, x ^ i / ↑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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | refine sum_le_sum_of_subset_of_nonneg (range_mono hj) fun _ _ _ ↦ ?_ | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x :=
calc
∑ i in range n, x ^ i / i ! ≤ lim (⟨_, isCauSeq_re (exp' x)⟩ : CauSeq ℝ Abs.abs) := by
refine' le_lim (CauSeq.le_of_exists ⟨n, fun j hj => _⟩)
simp only [exp', const_apply, re_sum]
norm_cast
... | Mathlib.Data.Complex.Exponential.1473_0.1bixbwhfBeJKySp | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x | Mathlib_Data_Complex_Exponential |
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