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 |
|---|---|---|---|---|---|---|
z : ℂ
n : ℕ
hn : abs z < ↑n
hn0 : 0 < ↑n
⊢ abs z / ↑n < 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... | rwa [div_lt_iff hn0, one_mul] | theorem isCauSeq_abs_exp (z : ℂ) :
IsCauSeq Abs.abs fun n => ∑ m in range n, abs (z ^ m / m.factorial) :=
let ⟨n, hn⟩ := exists_nat_gt (abs z)
have hn0 : (0 : ℝ) < n := lt_of_le_of_lt (abs.nonneg _) hn
series_ratio_test n (abs z / n) (div_nonneg (abs.nonneg _) (le_of_lt hn0))
(by | Mathlib.Data.Complex.Exponential.352_0.1bixbwhfBeJKySp | theorem isCauSeq_abs_exp (z : ℂ) :
IsCauSeq Abs.abs fun n => ∑ m in range n, abs (z ^ m / m.factorial) | Mathlib_Data_Complex_Exponential |
z : ℂ
n : ℕ
hn : abs z < ↑n
hn0 : 0 < ↑n
m : ℕ
hm : n ≤ m
⊢ abs' (abs (z ^ Nat.succ m / ↑(Nat.factorial (Nat.succ m)))) ≤ abs z / ↑n * abs' (abs (z ^ m / ↑(Nat.factorial m))) | /-
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 [abs_abs, abs_abs, Nat.factorial_succ, pow_succ, mul_comm m.succ, Nat.cast_mul, ← div_div,
mul_div_assoc, mul_div_right_comm, map_mul, map_div₀, abs_natCast] | theorem isCauSeq_abs_exp (z : ℂ) :
IsCauSeq Abs.abs fun n => ∑ m in range n, abs (z ^ m / m.factorial) :=
let ⟨n, hn⟩ := exists_nat_gt (abs z)
have hn0 : (0 : ℝ) < n := lt_of_le_of_lt (abs.nonneg _) hn
series_ratio_test n (abs z / n) (div_nonneg (abs.nonneg _) (le_of_lt hn0))
(by rwa [div_lt_iff hn0, one_... | Mathlib.Data.Complex.Exponential.352_0.1bixbwhfBeJKySp | theorem isCauSeq_abs_exp (z : ℂ) :
IsCauSeq Abs.abs fun n => ∑ m in range n, abs (z ^ m / m.factorial) | Mathlib_Data_Complex_Exponential |
z : ℂ
n : ℕ
hn : abs z < ↑n
hn0 : 0 < ↑n
m : ℕ
hm : n ≤ m
⊢ abs z / ↑(Nat.succ m) * abs (z ^ m / ↑(Nat.factorial m)) ≤ abs z / ↑n * abs (z ^ m / ↑(Nat.factorial m)) | /-
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... | gcongr | theorem isCauSeq_abs_exp (z : ℂ) :
IsCauSeq Abs.abs fun n => ∑ m in range n, abs (z ^ m / m.factorial) :=
let ⟨n, hn⟩ := exists_nat_gt (abs z)
have hn0 : (0 : ℝ) < n := lt_of_le_of_lt (abs.nonneg _) hn
series_ratio_test n (abs z / n) (div_nonneg (abs.nonneg _) (le_of_lt hn0))
(by rwa [div_lt_iff hn0, one_... | Mathlib.Data.Complex.Exponential.352_0.1bixbwhfBeJKySp | theorem isCauSeq_abs_exp (z : ℂ) :
IsCauSeq Abs.abs fun n => ∑ m in range n, abs (z ^ m / m.factorial) | Mathlib_Data_Complex_Exponential |
case h.h.h
z : ℂ
n : ℕ
hn : abs z < ↑n
hn0 : 0 < ↑n
m : ℕ
hm : n ≤ m
⊢ n ≤ Nat.succ m | /-
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_trans hm (Nat.le_succ _) | theorem isCauSeq_abs_exp (z : ℂ) :
IsCauSeq Abs.abs fun n => ∑ m in range n, abs (z ^ m / m.factorial) :=
let ⟨n, hn⟩ := exists_nat_gt (abs z)
have hn0 : (0 : ℝ) < n := lt_of_le_of_lt (abs.nonneg _) hn
series_ratio_test n (abs z / n) (div_nonneg (abs.nonneg _) (le_of_lt hn0))
(by rwa [div_lt_iff hn0, one_... | Mathlib.Data.Complex.Exponential.352_0.1bixbwhfBeJKySp | theorem isCauSeq_abs_exp (z : ℂ) :
IsCauSeq Abs.abs fun n => ∑ m in range n, abs (z ^ m / m.factorial) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cexp 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... | rw [exp] | @[simp]
theorem exp_zero : exp 0 = 1 := by
| Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ CauSeq.lim (exp' 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... | refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩ | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
| Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
ε : ℝ
ε0 : ε > 0
j : ℕ
hj : j ≥ 1
⊢ abs (↑(exp' 0 - const (⇑abs) 1) j) < ε | /-
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... | convert (config := .unfoldSameFun) ε0 | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
| Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
case h.e'_3
x y : ℂ
ε : ℝ
ε0 : ε > 0
j : ℕ
hj : j ≥ 1
⊢ abs (↑(exp' 0 - const (⇑abs) 1) j) = 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... | cases' j with j j | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
| Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
case h.e'_3.zero
x y : ℂ
ε : ℝ
ε0 : ε > 0
hj : Nat.zero ≥ 1
⊢ abs (↑(exp' 0 - const (⇑abs) 1) Nat.zero) = 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... | exact absurd hj (not_le_of_gt zero_lt_one) | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
cases' j with j j
· | Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
case h.e'_3.succ
x y : ℂ
ε : ℝ
ε0 : ε > 0
j : ℕ
hj : Nat.succ j ≥ 1
⊢ abs (↑(exp' 0 - const (⇑abs) 1) (Nat.succ j)) = 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... | dsimp [exp'] | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
cases' j with j j
· exact absurd hj (not_le_of_gt zero_lt_one)
· | Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
case h.e'_3.succ
x y : ℂ
ε : ℝ
ε0 : ε > 0
j : ℕ
hj : Nat.succ j ≥ 1
⊢ abs (∑ m in range (Nat.succ j), 0 ^ m / ↑(Nat.factorial m) - 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | induction' j with j ih | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
cases' j with j j
· exact absurd hj (not_le_of_gt zero_lt_one)
· dsimp [exp']
| Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
case h.e'_3.succ.zero
x y : ℂ
ε : ℝ
ε0 : ε > 0
hj : Nat.succ Nat.zero ≥ 1
⊢ abs (∑ m in range (Nat.succ Nat.zero), 0 ^ m / ↑(Nat.factorial m) - 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | dsimp [exp'] | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
cases' j with j j
· exact absurd hj (not_le_of_gt zero_lt_one)
· dsimp [exp']
induction' j with j ih
· | Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
case h.e'_3.succ.zero
x y : ℂ
ε : ℝ
ε0 : ε > 0
hj : Nat.succ Nat.zero ≥ 1
⊢ abs (∑ m in {0}, 0 ^ m / ↑(Nat.factorial m) - 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [show Nat.succ 0 = 1 from rfl] | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
cases' j with j j
· exact absurd hj (not_le_of_gt zero_lt_one)
· dsimp [exp']
induction' j with j ih
· ... | Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
case h.e'_3.succ.succ
x y : ℂ
ε : ℝ
ε0 : ε > 0
j : ℕ
ih : Nat.succ j ≥ 1 → abs (∑ m in range (Nat.succ j), 0 ^ m / ↑(Nat.factorial m) - 1) = 0
hj : Nat.succ (Nat.succ j) ≥ 1
⊢ abs (∑ m in range (Nat.succ (Nat.succ j)), 0 ^ m / ↑(Nat.factorial m) - 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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ih (by simp [show 1 = Nat.succ 0 from rfl, Nat.succ_le_succ])] | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
cases' j with j j
· exact absurd hj (not_le_of_gt zero_lt_one)
· dsimp [exp']
induction' j with j ih
· ... | Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
ε : ℝ
ε0 : ε > 0
j : ℕ
ih : Nat.succ j ≥ 1 → abs (∑ m in range (Nat.succ j), 0 ^ m / ↑(Nat.factorial m) - 1) = 0
hj : Nat.succ (Nat.succ j) ≥ 1
⊢ Nat.succ j ≥ 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 [show 1 = Nat.succ 0 from rfl, Nat.succ_le_succ] | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
cases' j with j j
· exact absurd hj (not_le_of_gt zero_lt_one)
· dsimp [exp']
induction' j with j ih
· ... | Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
case h.e'_3.succ.succ
x y : ℂ
ε : ℝ
ε0 : ε > 0
j : ℕ
ih : Nat.succ j ≥ 1 → abs (∑ m in range (Nat.succ j), 0 ^ m / ↑(Nat.factorial m) - 1) = 0
hj : Nat.succ (Nat.succ j) ≥ 1
⊢ abs (∑ m in range (Nat.succ (Nat.succ j)), 0 ^ m / ↑(Nat.factorial m) - 1) =
abs (∑ m in range (Nat.succ j), 0 ^ m / ↑(Nat.factorial m) - 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 [sum_range_succ, pow_succ] | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
cases' j with j j
· exact absurd hj (not_le_of_gt zero_lt_one)
· dsimp [exp']
induction' j with j ih
· ... | Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
case h.e'_3.succ.succ
x y : ℂ
ε : ℝ
ε0 : ε > 0
j : ℕ
ih : Nat.succ j ≥ 1 → abs (∑ m in range (Nat.succ j), 0 ^ m / ↑(Nat.factorial m) - 1) = 0
hj : Nat.succ (Nat.succ j) ≥ 1
⊢ abs
(∑ m in range j, 0 ^ m / ↑(Nat.factorial m) + 0 ^ j / ↑(Nat.factorial j) + 0 * 0 ^ j / ↑(Nat.factorial (j + 1)) -
1) =
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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | @[simp]
theorem exp_zero : exp 0 = 1 := by
rw [exp]
refine' lim_eq_of_equiv_const fun ε ε0 => ⟨1, fun j hj => _⟩
convert (config := .unfoldSameFun) ε0 -- porting note: ε0 : ε > 0 but goal is _ < ε
cases' j with j j
· exact absurd hj (not_le_of_gt zero_lt_one)
· dsimp [exp']
induction' j with j ih
· ... | Mathlib.Data.Complex.Exponential.487_0.1bixbwhfBeJKySp | @[simp]
theorem exp_zero : exp 0 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cexp (x + y) = cexp x * cexp 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... | have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_mul_inv, sum_mul]
refine' Finset.sum_congr rfl fun ... | theorem exp_add : exp (x + y) = exp x * exp y := by
| Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ ∀ (j : ℕ),
∑ m in range j, (x + y) ^ m / ↑(Nat.factorial m) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / ↑(Nat.factorial k) * (y ^ (i - k) / ↑(Nat.factorial (i - k))) | /-
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... | intro j | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
| Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
j : ℕ
⊢ ∑ m in range j, (x + y) ^ m / ↑(Nat.factorial m) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / ↑(Nat.factorial k) * (y ^ (i - k) / ↑(Nat.factorial (i - k))) | /-
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' Finset.sum_congr rfl fun m _ => _ | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
| Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
j m : ℕ
x✝ : m ∈ range j
⊢ (x + y) ^ m / ↑(Nat.factorial m) =
∑ k in range (m + 1), x ^ k / ↑(Nat.factorial k) * (y ^ (m - k) / ↑(Nat.factorial (m - k))) | /-
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_pow, div_eq_mul_inv, sum_mul] | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
| Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
j m : ℕ
x✝ : m ∈ range j
⊢ ∑ x_1 in range (m + 1), x ^ x_1 * y ^ (m - x_1) * ↑(Nat.choose m x_1) * (↑(Nat.factorial m))⁻¹ =
∑ k in range (m + 1), x ^ k / ↑(Nat.factorial k) * (y ^ (m - k) / ↑(Nat.factorial (m - k))) | /-
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' Finset.sum_congr rfl fun I hi => _ | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
j m : ℕ
x✝ : m ∈ range j
I : ℕ
hi : I ∈ range (m + 1)
⊢ x ^ I * y ^ (m - I) * ↑(Nat.choose m I) * (↑(Nat.factorial m))⁻¹ =
x ^ I / ↑(Nat.factorial I) * (y ^ (m - I) / ↑(Nat.factorial (m - 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... | have h₁ : (m.choose I : ℂ) ≠ 0 :=
Nat.cast_ne_zero.2 (pos_iff_ne_zero.1 (Nat.choose_pos (Nat.le_of_lt_succ (mem_range.1 hi)))) | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
j m : ℕ
x✝ : m ∈ range j
I : ℕ
hi : I ∈ range (m + 1)
h₁ : ↑(Nat.choose m I) ≠ 0
⊢ x ^ I * y ^ (m - I) * ↑(Nat.choose m I) * (↑(Nat.factorial m))⁻¹ =
x ^ I / ↑(Nat.factorial I) * (y ^ (m - I) / ↑(Nat.factorial (m - 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... | have h₂ := Nat.choose_mul_factorial_mul_factorial (Nat.le_of_lt_succ <| Finset.mem_range.1 hi) | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
j m : ℕ
x✝ : m ∈ range j
I : ℕ
hi : I ∈ range (m + 1)
h₁ : ↑(Nat.choose m I) ≠ 0
h₂ : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m
⊢ x ^ I * y ^ (m - I) * ↑(Nat.choose m I) * (↑(Nat.factorial m))⁻¹ =
x ^ I / ↑(Nat.factorial I) * (y ^ (m - I) / ↑(Nat.factorial (m - 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 [← h₂, Nat.cast_mul, Nat.cast_mul, mul_inv, mul_inv] | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
j m : ℕ
x✝ : m ∈ range j
I : ℕ
hi : I ∈ range (m + 1)
h₁ : ↑(Nat.choose m I) ≠ 0
h₂ : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m
⊢ x ^ I * y ^ (m - I) * ↑(Nat.choose m I) *
((↑(Nat.choose m I))⁻¹ * (↑(Nat.factorial I))⁻¹ * (↑(Nat.factorial (m - I)))⁻¹) =
x ^ I / ↑(Nat.f... | /-
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 [mul_left_comm (m.choose I : ℂ), mul_assoc, mul_left_comm (m.choose I : ℂ)⁻¹,
mul_comm (m.choose I : ℂ)] | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
j m : ℕ
x✝ : m ∈ range j
I : ℕ
hi : I ∈ range (m + 1)
h₁ : ↑(Nat.choose m I) ≠ 0
h₂ : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m
⊢ x ^ I *
(y ^ (m - I) *
((↑(Nat.factorial I))⁻¹ * ((↑(Nat.factorial (m - I)))⁻¹ * ((↑(Nat.choose m I))⁻¹ * ↑(Nat.choose m I))))) =
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 [inv_mul_cancel h₁] | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
j m : ℕ
x✝ : m ∈ range j
I : ℕ
hi : I ∈ range (m + 1)
h₁ : ↑(Nat.choose m I) ≠ 0
h₂ : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m
⊢ x ^ I * (y ^ (m - I) * ((↑(Nat.factorial I))⁻¹ * ((↑(Nat.factorial (m - I)))⁻¹ * 1))) =
x ^ I / ↑(Nat.factorial I) * (y ^ (m - I) / ↑(Nat.factori... | /-
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 [div_eq_mul_inv, mul_comm, mul_assoc, mul_left_comm] | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
hj :
∀ (j : ℕ),
∑ m in range j, (x + y) ^ m / ↑(Nat.factorial m) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / ↑(Nat.factorial k) * (y ^ (i - k) / ↑(Nat.factorial (i - k)))
⊢ cexp (x + y) = cexp x * cexp 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_rw [exp, exp', lim_mul_lim] | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
hj :
∀ (j : ℕ),
∑ m in range j, (x + y) ^ m / ↑(Nat.factorial m) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / ↑(Nat.factorial k) * (y ^ (i - k) / ↑(Nat.factorial (i - k)))
⊢ CauSeq.lim
{ val := fun n => ∑ m in range n, (x + y) ^ m / ↑(Nat.factorial m),
property := (_ : IsCauSeq ⇑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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | apply (lim_eq_lim_of_equiv _).symm | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
hj :
∀ (j : ℕ),
∑ m in range j, (x + y) ^ m / ↑(Nat.factorial m) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / ↑(Nat.factorial k) * (y ^ (i - k) / ↑(Nat.factorial (i - k)))
⊢ { val := fun n => ∑ m in range n, x ^ m / ↑(Nat.factorial m),
property := (_ : IsCauSeq ⇑abs fun n => ∑ m in range ... | /-
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 [hj] | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
hj :
∀ (j : ℕ),
∑ m in range j, (x + y) ^ m / ↑(Nat.factorial m) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / ↑(Nat.factorial k) * (y ^ (i - k) / ↑(Nat.factorial (i - k)))
⊢ { val := fun n => ∑ m in range n, x ^ m / ↑(Nat.factorial m),
property := (_ : IsCauSeq ⇑abs fun n => ∑ m in range ... | /-
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 cauchy_product (isCauSeq_abs_exp x) (isCauSeq_exp y) | theorem exp_add : exp (x + y) = exp x * exp y := by
have hj : ∀ j : ℕ, (∑ m in range j, (x + y) ^ m / m.factorial) =
∑ i in range j, ∑ k in range (i + 1), x ^ k / k.factorial *
(y ^ (i - k) / (i - k).factorial) := by
intro j
refine' Finset.sum_congr rfl fun m _ => _
rw [add_pow, div_eq_m... | Mathlib.Data.Complex.Exponential.502_0.1bixbwhfBeJKySp | theorem exp_add : exp (x + y) = exp x * exp y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ (fun z => cexp (Multiplicative.toAdd z)) 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 ℂ` to `ℂ` -/
noncomputable def expMonoidHom : MonoidHom (Multiplicative ℂ) ℂ :=
{ toFun := fun z => exp (Multiplicative.toAdd z),
map_one' := by | Mathlib.Data.Complex.Exponential.525_0.1bixbwhfBeJKySp | /-- the exponential function as a monoid hom from `Multiplicative ℂ` to `ℂ` -/
noncomputable def expMonoidHom : MonoidHom (Multiplicative ℂ) ℂ | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ ∀ (x y : Multiplicative ℂ),
OneHom.toFun { toFun := fun z => cexp (Multiplicative.toAdd z), map_one' := (_ : cexp 0 = 1) } (x * y) =
OneHom.toFun { toFun := fun z => cexp (Multiplicative.toAdd z), map_one' := (_ : cexp 0 = 1) } x *
OneHom.toFun { toFun := fun z => cexp (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 ℂ` to `ℂ` -/
noncomputable def expMonoidHom : MonoidHom (Multiplicative ℂ) ℂ :=
{ toFun := fun z => exp (Multiplicative.toAdd z),
map_one' := by simp,
map_mul' := by | Mathlib.Data.Complex.Exponential.525_0.1bixbwhfBeJKySp | /-- the exponential function as a monoid hom from `Multiplicative ℂ` to `ℂ` -/
noncomputable def expMonoidHom : MonoidHom (Multiplicative ℂ) ℂ | Mathlib_Data_Complex_Exponential |
x✝ y x : ℂ
⊢ cexp (↑0 * x) = 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 [Nat.cast_zero, zero_mul, exp_zero, pow_zero] | theorem exp_nat_mul (x : ℂ) : ∀ n : ℕ, exp (n * x) = exp x ^ n
| 0 => by | Mathlib.Data.Complex.Exponential.547_0.1bixbwhfBeJKySp | theorem exp_nat_mul (x : ℂ) : ∀ n : ℕ, exp (n * x) = exp x ^ n
| 0 => by rw [Nat.cast_zero, zero_mul, exp_zero, pow_zero]
| Nat.succ n => by rw [pow_succ', Nat.cast_add_one, add_mul, exp_add, ← exp_nat_mul _ n, one_mul] | Mathlib_Data_Complex_Exponential |
x✝ y x : ℂ
n : ℕ
⊢ cexp (↑(Nat.succ n) * x) = cexp x ^ Nat.succ 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... | rw [pow_succ', Nat.cast_add_one, add_mul, exp_add, ← exp_nat_mul _ n, one_mul] | theorem exp_nat_mul (x : ℂ) : ∀ n : ℕ, exp (n * x) = exp x ^ n
| 0 => by rw [Nat.cast_zero, zero_mul, exp_zero, pow_zero]
| Nat.succ n => by | Mathlib.Data.Complex.Exponential.547_0.1bixbwhfBeJKySp | theorem exp_nat_mul (x : ℂ) : ∀ n : ℕ, exp (n * x) = exp x ^ n
| 0 => by rw [Nat.cast_zero, zero_mul, exp_zero, pow_zero]
| Nat.succ n => by rw [pow_succ', Nat.cast_add_one, add_mul, exp_add, ← exp_nat_mul _ n, one_mul] | Mathlib_Data_Complex_Exponential |
x y : ℂ
h : cexp x = 0
⊢ 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... | rw [← exp_zero, ← add_neg_self x, exp_add, h] | theorem exp_ne_zero : exp x ≠ 0 := fun h =>
zero_ne_one <| by | Mathlib.Data.Complex.Exponential.552_0.1bixbwhfBeJKySp | theorem exp_ne_zero : exp x ≠ 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
h : cexp x = 0
⊢ x + -x = 0 * cexp (-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 | theorem exp_ne_zero : exp x ≠ 0 := fun h =>
zero_ne_one <| by rw [← exp_zero, ← add_neg_self x, exp_add, h]; | Mathlib.Data.Complex.Exponential.552_0.1bixbwhfBeJKySp | theorem exp_ne_zero : exp x ≠ 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cexp (-x) = (cexp 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... | rw [← mul_right_inj' (exp_ne_zero x), ← exp_add] | theorem exp_neg : exp (-x) = (exp x)⁻¹ := by
| Mathlib.Data.Complex.Exponential.556_0.1bixbwhfBeJKySp | theorem exp_neg : exp (-x) = (exp x)⁻¹ | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cexp (x + -x) = cexp x * (cexp 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 [mul_inv_cancel (exp_ne_zero x)] | theorem exp_neg : exp (-x) = (exp x)⁻¹ := by
rw [← mul_right_inj' (exp_ne_zero x), ← exp_add]; | Mathlib.Data.Complex.Exponential.556_0.1bixbwhfBeJKySp | theorem exp_neg : exp (-x) = (exp x)⁻¹ | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cexp (x - y) = cexp x / cexp 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.560_0.1bixbwhfBeJKySp | theorem exp_sub : exp (x - y) = exp x / exp y | Mathlib_Data_Complex_Exponential |
x y z : ℂ
n : ℤ
⊢ cexp (↑n * z) = cexp z ^ 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... | cases n | theorem exp_int_mul (z : ℂ) (n : ℤ) : Complex.exp (n * z) = Complex.exp z ^ n := by
| Mathlib.Data.Complex.Exponential.564_0.1bixbwhfBeJKySp | theorem exp_int_mul (z : ℂ) (n : ℤ) : Complex.exp (n * z) = Complex.exp z ^ n | Mathlib_Data_Complex_Exponential |
case ofNat
x y z : ℂ
a✝ : ℕ
⊢ cexp (↑(Int.ofNat a✝) * z) = cexp z ^ Int.ofNat a✝ | /-
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] | theorem exp_int_mul (z : ℂ) (n : ℤ) : Complex.exp (n * z) = Complex.exp z ^ n := by
cases n
· | Mathlib.Data.Complex.Exponential.564_0.1bixbwhfBeJKySp | theorem exp_int_mul (z : ℂ) (n : ℤ) : Complex.exp (n * z) = Complex.exp z ^ n | Mathlib_Data_Complex_Exponential |
case negSucc
x y z : ℂ
a✝ : ℕ
⊢ cexp (↑(Int.negSucc a✝) * z) = cexp z ^ Int.negSucc a✝ | /-
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, add_mul, pow_add, exp_neg, exp_nat_mul] | theorem exp_int_mul (z : ℂ) (n : ℤ) : Complex.exp (n * z) = Complex.exp z ^ n := by
cases n
· simp [exp_nat_mul]
· | Mathlib.Data.Complex.Exponential.564_0.1bixbwhfBeJKySp | theorem exp_int_mul (z : ℂ) (n : ℤ) : Complex.exp (n * z) = Complex.exp z ^ n | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cexp ((starRingEnd ℂ) x) = (starRingEnd ℂ) (cexp 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... | dsimp [exp] | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) := by
| Mathlib.Data.Complex.Exponential.570_0.1bixbwhfBeJKySp | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ CauSeq.lim (exp' ((starRingEnd ℂ) x)) = (starRingEnd ℂ) (CauSeq.lim (exp' 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 [← lim_conj] | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) := by
dsimp [exp]
| Mathlib.Data.Complex.Exponential.570_0.1bixbwhfBeJKySp | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ CauSeq.lim (exp' ((starRingEnd ℂ) x)) = CauSeq.lim (cauSeqConj (exp' 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... | refine' congr_arg CauSeq.lim (CauSeq.ext fun _ => _) | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) := by
dsimp [exp]
rw [← lim_conj]
| Mathlib.Data.Complex.Exponential.570_0.1bixbwhfBeJKySp | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
x✝ : ℕ
⊢ ↑(exp' ((starRingEnd ℂ) x)) x✝ = ↑(cauSeqConj (exp' x)) 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... | dsimp [exp', Function.comp_def, isCauSeq_conj, cauSeqConj] | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) := by
dsimp [exp]
rw [← lim_conj]
refine' congr_arg CauSeq.lim (CauSeq.ext fun _ => _)
| Mathlib.Data.Complex.Exponential.570_0.1bixbwhfBeJKySp | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
x✝ : ℕ
⊢ ∑ m in range x✝, (starRingEnd ℂ) x ^ m / ↑(Nat.factorial m) =
(starRingEnd ℂ) (∑ m in range x✝, x ^ m / ↑(Nat.factorial m)) | /-
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 [(starRingEnd _).map_sum] | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) := by
dsimp [exp]
rw [← lim_conj]
refine' congr_arg CauSeq.lim (CauSeq.ext fun _ => _)
dsimp [exp', Function.comp_def, isCauSeq_conj, cauSeqConj]
| Mathlib.Data.Complex.Exponential.570_0.1bixbwhfBeJKySp | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
x✝ : ℕ
⊢ ∑ m in range x✝, (starRingEnd ℂ) x ^ m / ↑(Nat.factorial m) =
∑ x_1 in range x✝, (starRingEnd ℂ) (x ^ x_1 / ↑(Nat.factorial 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... | refine' sum_congr rfl fun n _ => _ | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) := by
dsimp [exp]
rw [← lim_conj]
refine' congr_arg CauSeq.lim (CauSeq.ext fun _ => _)
dsimp [exp', Function.comp_def, isCauSeq_conj, cauSeqConj]
rw [(starRingEnd _).map_sum]
| Mathlib.Data.Complex.Exponential.570_0.1bixbwhfBeJKySp | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
x✝¹ n : ℕ
x✝ : n ∈ range x✝¹
⊢ (starRingEnd ℂ) x ^ n / ↑(Nat.factorial n) = (starRingEnd ℂ) (x ^ n / ↑(Nat.factorial 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... | rw [map_div₀, map_pow, ← ofReal_nat_cast, conj_ofReal] | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) := by
dsimp [exp]
rw [← lim_conj]
refine' congr_arg CauSeq.lim (CauSeq.ext fun _ => _)
dsimp [exp', Function.comp_def, isCauSeq_conj, cauSeqConj]
rw [(starRingEnd _).map_sum]
refine' sum_congr rfl fun n _ => _
| Mathlib.Data.Complex.Exponential.570_0.1bixbwhfBeJKySp | @[simp]
theorem exp_conj : exp (conj x) = conj (exp x) | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (starRingEnd ℂ) (cexp ↑x) = cexp ↑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 [← exp_conj, conj_ofReal] | @[simp]
theorem ofReal_exp_ofReal_re (x : ℝ) : ((exp x).re : ℂ) = exp x :=
conj_eq_iff_re.1 <| by | Mathlib.Data.Complex.Exponential.581_0.1bixbwhfBeJKySp | @[simp]
theorem ofReal_exp_ofReal_re (x : ℝ) : ((exp x).re : ℂ) = exp x | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (cexp ↑x).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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_exp_ofReal_re, ofReal_im] | @[simp]
theorem exp_ofReal_im (x : ℝ) : (exp x).im = 0 := by | Mathlib.Data.Complex.Exponential.591_0.1bixbwhfBeJKySp | @[simp]
theorem exp_ofReal_im (x : ℝ) : (exp x).im = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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.607_0.1bixbwhfBeJKySp | @[simp]
theorem sinh_zero : sinh 0 = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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.611_0.1bixbwhfBeJKySp | @[simp]
theorem sinh_neg : sinh (-x) = -sinh x | Mathlib_Data_Complex_Exponential |
x y a b c d : ℂ
⊢ (a - b) * (c + d) + (a + b) * (c - d) = 2 * (a * c - b * d) | /-
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... | ring | private theorem sinh_add_aux {a b c d : ℂ} :
(a - b) * (c + d) + (a + b) * (c - d) = 2 * (a * c - b * d) := by | Mathlib.Data.Complex.Exponential.615_0.1bixbwhfBeJKySp | private theorem sinh_add_aux {a b c d : ℂ} :
(a - b) * (c + d) + (a + b) * (c - d) = 2 * (a * c - b * d) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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 [← mul_right_inj' (two_ne_zero' ℂ), two_sinh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
mul_assoc, two_sinh, mul_left_comm, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
mul_left_comm, two_cosh, ← mul_assoc, two_cosh] | theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y := by
| Mathlib.Data.Complex.Exponential.618_0.1bixbwhfBeJKySp | theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ (cexp x - cexp (-x)) * (cexp y + cexp (-y)) + (cexp x + cexp (-x)) * (cexp y - cexp (-y)) =
2 * (cexp x * cexp y - cexp (-x) * cexp (-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... | exact sinh_add_aux | theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y := by
rw [← mul_right_inj' (two_ne_zero' ℂ), two_sinh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
mul_assoc, two_sinh, mul_left_comm, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
mul_left_comm, two_cosh, ← mul_assoc, two_cosh]
| Mathlib.Data.Complex.Exponential.618_0.1bixbwhfBeJKySp | theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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.625_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_zero : cosh 0 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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 [add_comm, cosh, exp_neg] | @[simp]
theorem cosh_neg : cosh (-x) = cosh x := by | Mathlib.Data.Complex.Exponential.629_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_neg : cosh (-x) = cosh x | Mathlib_Data_Complex_Exponential |
x y a b c d : ℂ
⊢ (a + b) * (c + d) + (a - b) * (c - d) = 2 * (a * c + b * d) | /-
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... | ring | private theorem cosh_add_aux {a b c d : ℂ} :
(a + b) * (c + d) + (a - b) * (c - d) = 2 * (a * c + b * d) := by | Mathlib.Data.Complex.Exponential.633_0.1bixbwhfBeJKySp | private theorem cosh_add_aux {a b c d : ℂ} :
(a + b) * (c + d) + (a - b) * (c - d) = 2 * (a * c + b * d) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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 [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
mul_assoc, two_cosh, ← mul_assoc, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
mul_left_comm, two_cosh, mul_left_comm, two_sinh] | theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y := by
| Mathlib.Data.Complex.Exponential.636_0.1bixbwhfBeJKySp | theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ (cexp x + cexp (-x)) * (cexp y + cexp (-y)) + (cexp x - cexp (-x)) * (cexp y - cexp (-y)) =
2 * (cexp x * cexp y + cexp (-x) * cexp (-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... | exact cosh_add_aux | theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y := by
rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
mul_assoc, two_cosh, ← mul_assoc, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
mul_left_comm, two_cosh, mul_left_comm, two_sinh]
| Mathlib.Data.Complex.Exponential.636_0.1bixbwhfBeJKySp | theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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.643_0.1bixbwhfBeJKySp | theorem sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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.647_0.1bixbwhfBeJKySp | theorem cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sinh ((starRingEnd ℂ) x) = (starRingEnd ℂ) (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 [sinh, ← RingHom.map_neg, exp_conj, exp_conj, ← RingHom.map_sub, sinh, map_div₀] | theorem sinh_conj : sinh (conj x) = conj (sinh x) := by
| Mathlib.Data.Complex.Exponential.651_0.1bixbwhfBeJKySp | theorem sinh_conj : sinh (conj x) = conj (sinh x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ (starRingEnd ℂ) (cexp x - cexp (-x)) / 2 = (starRingEnd ℂ) (cexp x - cexp (-x)) / (starRingEnd ℂ) 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 [← one_add_one_eq_two] | theorem sinh_conj : sinh (conj x) = conj (sinh x) := by
rw [sinh, ← RingHom.map_neg, exp_conj, exp_conj, ← RingHom.map_sub, sinh, map_div₀]
--Porting note: not nice
| Mathlib.Data.Complex.Exponential.651_0.1bixbwhfBeJKySp | theorem sinh_conj : sinh (conj x) = conj (sinh x) | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (starRingEnd ℂ) (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... | rw [← sinh_conj, conj_ofReal] | @[simp]
theorem ofReal_sinh_ofReal_re (x : ℝ) : ((sinh x).re : ℂ) = sinh x :=
conj_eq_iff_re.1 <| by | Mathlib.Data.Complex.Exponential.657_0.1bixbwhfBeJKySp | @[simp]
theorem ofReal_sinh_ofReal_re (x : ℝ) : ((sinh x).re : ℂ) = sinh x | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (sinh ↑x).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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_sinh_ofReal_re, ofReal_im] | @[simp]
theorem sinh_ofReal_im (x : ℝ) : (sinh x).im = 0 := by | Mathlib.Data.Complex.Exponential.667_0.1bixbwhfBeJKySp | @[simp]
theorem sinh_ofReal_im (x : ℝ) : (sinh x).im = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cosh ((starRingEnd ℂ) x) = (starRingEnd ℂ) (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 [cosh, ← RingHom.map_neg, exp_conj, exp_conj, ← RingHom.map_add, cosh, map_div₀] | theorem cosh_conj : cosh (conj x) = conj (cosh x) := by
| Mathlib.Data.Complex.Exponential.675_0.1bixbwhfBeJKySp | theorem cosh_conj : cosh (conj x) = conj (cosh x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ (starRingEnd ℂ) (cexp x + cexp (-x)) / 2 = (starRingEnd ℂ) (cexp x + cexp (-x)) / (starRingEnd ℂ) 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 [← one_add_one_eq_two] | theorem cosh_conj : cosh (conj x) = conj (cosh x) := by
rw [cosh, ← RingHom.map_neg, exp_conj, exp_conj, ← RingHom.map_add, cosh, map_div₀]
--Porting note: not nice
| Mathlib.Data.Complex.Exponential.675_0.1bixbwhfBeJKySp | theorem cosh_conj : cosh (conj x) = conj (cosh x) | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (starRingEnd ℂ) (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... | rw [← cosh_conj, conj_ofReal] | theorem ofReal_cosh_ofReal_re (x : ℝ) : ((cosh x).re : ℂ) = cosh x :=
conj_eq_iff_re.1 <| by | Mathlib.Data.Complex.Exponential.681_0.1bixbwhfBeJKySp | theorem ofReal_cosh_ofReal_re (x : ℝ) : ((cosh x).re : ℂ) = cosh x | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (cosh ↑x).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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_cosh_ofReal_re, ofReal_im] | @[simp]
theorem cosh_ofReal_im (x : ℝ) : (cosh x).im = 0 := by | Mathlib.Data.Complex.Exponential.690_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_ofReal_im (x : ℝ) : (cosh x).im = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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.703_0.1bixbwhfBeJKySp | @[simp]
theorem tanh_zero : tanh 0 = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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.707_0.1bixbwhfBeJKySp | @[simp]
theorem tanh_neg : tanh (-x) = -tanh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ tanh ((starRingEnd ℂ) x) = (starRingEnd ℂ) (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... | rw [tanh, sinh_conj, cosh_conj, ← map_div₀, tanh] | theorem tanh_conj : tanh (conj x) = conj (tanh x) := by
| Mathlib.Data.Complex.Exponential.711_0.1bixbwhfBeJKySp | theorem tanh_conj : tanh (conj x) = conj (tanh x) | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (starRingEnd ℂ) (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... | rw [← tanh_conj, conj_ofReal] | @[simp]
theorem ofReal_tanh_ofReal_re (x : ℝ) : ((tanh x).re : ℂ) = tanh x :=
conj_eq_iff_re.1 <| by | Mathlib.Data.Complex.Exponential.715_0.1bixbwhfBeJKySp | @[simp]
theorem ofReal_tanh_ofReal_re (x : ℝ) : ((tanh x).re : ℂ) = tanh x | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (tanh ↑x).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
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_tanh_ofReal_re, ofReal_im] | @[simp]
theorem tanh_ofReal_im (x : ℝ) : (tanh x).im = 0 := by | Mathlib.Data.Complex.Exponential.725_0.1bixbwhfBeJKySp | @[simp]
theorem tanh_ofReal_im (x : ℝ) : (tanh x).im = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cosh x + sinh x = cexp 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 [← mul_right_inj' (two_ne_zero' ℂ), mul_add, two_cosh, two_sinh, add_add_sub_cancel, two_mul] | @[simp]
theorem cosh_add_sinh : cosh x + sinh x = exp x := by
| Mathlib.Data.Complex.Exponential.733_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_add_sinh : cosh x + sinh x = exp x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sinh x + cosh x = cexp 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.738_0.1bixbwhfBeJKySp | @[simp]
theorem sinh_add_cosh : sinh x + cosh x = exp x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cosh x - sinh x = cexp (-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 [← mul_right_inj' (two_ne_zero' ℂ), mul_sub, two_cosh, two_sinh, add_sub_sub_cancel, two_mul] | @[simp]
theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) := by
| Mathlib.Data.Complex.Exponential.752_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sinh x - cosh x = -cexp (-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.757_0.1bixbwhfBeJKySp | @[simp]
theorem sinh_sub_cosh : sinh x - cosh x = -exp (-x) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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 [sq_sub_sq, cosh_add_sinh, cosh_sub_sinh, ← exp_add, add_neg_self, exp_zero] | @[simp]
theorem cosh_sq_sub_sinh_sq : cosh x ^ 2 - sinh x ^ 2 = 1 := by
| Mathlib.Data.Complex.Exponential.761_0.1bixbwhfBeJKySp | @[simp]
theorem cosh_sq_sub_sinh_sq : cosh x ^ 2 - sinh x ^ 2 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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 [← cosh_sq_sub_sinh_sq x] | theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 := by
| Mathlib.Data.Complex.Exponential.766_0.1bixbwhfBeJKySp | theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cosh x ^ 2 = sinh x ^ 2 + (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... | ring | theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 := by
rw [← cosh_sq_sub_sinh_sq x]
| Mathlib.Data.Complex.Exponential.766_0.1bixbwhfBeJKySp | theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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 [← cosh_sq_sub_sinh_sq x] | theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by
| Mathlib.Data.Complex.Exponential.771_0.1bixbwhfBeJKySp | theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sinh x ^ 2 = cosh x ^ 2 - (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... | ring | theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by
rw [← cosh_sq_sub_sinh_sq x]
| Mathlib.Data.Complex.Exponential.771_0.1bixbwhfBeJKySp | theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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 [two_mul, cosh_add, sq, sq] | theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 := by | Mathlib.Data.Complex.Exponential.776_0.1bixbwhfBeJKySp | theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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 [two_mul, sinh_add] | theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x := by
| Mathlib.Data.Complex.Exponential.779_0.1bixbwhfBeJKySp | theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sinh x * cosh x + cosh x * sinh 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... | ring | theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x := by
rw [two_mul, sinh_add]
| Mathlib.Data.Complex.Exponential.779_0.1bixbwhfBeJKySp | theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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... | have h1 : x + 2 * x = 3 * x := by ring | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
| Mathlib.Data.Complex.Exponential.784_0.1bixbwhfBeJKySp | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ x + 2 * x = 3 * 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... | ring | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
have h1 : x + 2 * x = 3 * x := by | Mathlib.Data.Complex.Exponential.784_0.1bixbwhfBeJKySp | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ 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 [← h1, cosh_add x (2 * x)] | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
have h1 : x + 2 * x = 3 * x := by ring
| Mathlib.Data.Complex.Exponential.784_0.1bixbwhfBeJKySp | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cosh x * cosh (2 * x) + sinh x * sinh (2 * 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 only [cosh_two_mul, sinh_two_mul] | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cosh_add x (2 * x)]
| Mathlib.Data.Complex.Exponential.784_0.1bixbwhfBeJKySp | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cosh x * (cosh x ^ 2 + sinh x ^ 2) + sinh x * (2 * sinh x * cosh 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... | have h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2 := by ring | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cosh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
| Mathlib.Data.Complex.Exponential.784_0.1bixbwhfBeJKySp | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ sinh x * (2 * sinh x * cosh x) = 2 * cosh x * 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... | ring | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cosh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2 := by | Mathlib.Data.Complex.Exponential.784_0.1bixbwhfBeJKySp | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2
⊢ cosh x * (cosh x ^ 2 + sinh x ^ 2) + sinh x * (2 * sinh x * cosh 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 [h2, sinh_sq] | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cosh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2 := by ring
| Mathlib.Data.Complex.Exponential.784_0.1bixbwhfBeJKySp | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2
⊢ cosh x * (cosh x ^ 2 + (cosh x ^ 2 - 1)) + 2 * cosh x * (cosh x ^ 2 - 1) = 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... | ring | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cosh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2 := by ring
rw [h2, sinh_sq]
| Mathlib.Data.Complex.Exponential.784_0.1bixbwhfBeJKySp | theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ 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... | have h1 : x + 2 * x = 3 * x := by ring | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
| Mathlib.Data.Complex.Exponential.793_0.1bixbwhfBeJKySp | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ x + 2 * x = 3 * 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... | ring | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
have h1 : x + 2 * x = 3 * x := by | Mathlib.Data.Complex.Exponential.793_0.1bixbwhfBeJKySp | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
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