mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ 0 ≤ re z ∧ im z = 0 ↔ ∃ x ≥ 0, re ↑x = re z ∧ im ↑x = im z

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ 0 ≤ re z ∧ im z = 0 ↔ ∃ x ≥ 0, re ↑x = re z ∧ im ↑x = im z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ 0 < re z ∧ im z = 0 ↔ ∃ x > 0, re ↑x = re z ∧ im ↑x = im z

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ 0 < re z ∧ im z = 0 ↔ ∃ x > 0, re ↑x = re z ∧ im ↑x = im z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ re z ≤ 0 ∧ im z = 0 ↔ ∃ x ≤ 0, re ↑x = re z ∧ im ↑x = im z

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ re z ≤ 0 ∧ im z = 0 ↔ ∃ x ≤ 0, re ↑x = re z ∧ im ↑x = im z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ re z < 0 ∧ im z = 0 ↔ ∃ x < 0, re ↑x = re z ∧ im ↑x = im z

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ re z < 0 ∧ im z = 0 ↔ ∃ x < 0, re ↑x = re z ∧ im ↑x = im z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
x : K
⊢ ‖imLm x‖ ≤ 1 * ‖x‖

Tactic:
by
    rw [one_mul]
    exact abs_im_le_norm x

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x y : K
⊢ ‖re (x - y)‖ ≤ ‖x - y‖

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
x y : K
⊢ ‖re (x - y)‖ ≤ ‖x - y‖
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x y : K
⊢ ‖im (x - y)‖ ≤ ‖x - y‖

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
x y : K
⊢ ‖im (x - y)‖ ≤ ‖x - y‖
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝² : RCLike K
𝕜 : Type u_3
𝕜' : Type u_4
inst✝¹ : RCLike 𝕜
inst✝ : RCLike 𝕜'
x✝¹ x✝ : 𝕜
⊢ (algebraMap ℝ 𝕜') (re x✝¹) + (algebraMap ℝ 𝕜') (re x✝) +
      ((algebraMap ℝ 𝕜') (im x✝¹) * I + (algebraMap ℝ 𝕜') (im x✝) * I) =
    ↑(re x✝¹) + ↑(im x✝¹) * I + (↑(re x✝) + ↑(im x✝) * I)

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝² : RCLike K
𝕜 : Type u_3
𝕜' : Type u_4
inst✝¹ : RCLike 𝕜
inst✝ : RCLike 𝕜'
x✝¹ x✝ : 𝕜
⊢ (algebraMap ℝ 𝕜') (re x✝¹) + (algebraMap ℝ 𝕜') (re x✝) +
      ((algebraMap ℝ 𝕜') (im x✝¹) * I + (algebraMap ℝ 𝕜') (im x✝) * I) =
    ↑(re x✝¹) + ↑(im x✝¹) * I + (↑(re x✝) + ↑(im x✝) * I)
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x✝ : K
⊢ (map K K) x✝ = (ContinuousLinearMap.id ℝ K) x✝

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
x✝ : K
⊢ (map K K) x✝ = (ContinuousLinearMap.id ℝ K) x✝
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x✝ : K
⊢ (map K ℝ) x✝ = reCLM x✝

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
x✝ : K
⊢ (map K ℝ) x✝ = reCLM x✝
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x✝ : K
⊢ ↑(re x✝) = reCLM x✝

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
x✝ : K
⊢ ↑(re x✝) = reCLM x✝
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
⊢ (map ℝ K) 1 = ofRealCLM 1

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : RCLike K
⊢ (map ℝ K) 1 = ofRealCLM 1
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
h : I = 0
x : K
⊢ ↑(re x) = ↑(re x) + ↑(im x) * I

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
h : I = 0
x : K
⊢ ↑(re x) = ↑(re x) + ↑(im x) * I
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
h : I = 0
z : K
⊢ ‖{ toFun := (realRingEquiv h).toFun, map_add' := ⋯, map_smul' := ⋯, invFun := (realRingEquiv h).invFun, left_inv := ⋯,
          right_inv := ⋯ }
        (↑(re z) + ↑(im z) * I)‖ =
    ‖↑(re z) + ↑(im z) * I‖

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
h : I = 0
z : K
⊢ ‖{ toFun := (realRingEquiv h).toFun, map_add' := ⋯, map_smul' := ⋯, invFun := (realRingEquiv h).invFun, left_inv := ⋯,
          right_inv := ⋯ }
        (↑(re z) + ↑(im z) * I)‖ =
    ‖↑(re z) + ↑(im z) * I‖
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (x y : ℝ), 0 ≤ x → x < y → ∃ a, x < ‖a‖ ∧ ‖a‖ < y

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (x y : ℝ), 0 ≤ x → x < y → ∃ a, x < ‖a‖ ∧ ‖a‖ < y
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ Function.Involutive star

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ Function.Involutive star
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r s : 𝕜), star (r * s) = star s * star r

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r s : 𝕜), star (r * s) = star s * star r
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r s : 𝕜), star (r + s) = star r + star s

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r s : 𝕜), star (r + s) = star r + star s
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ (algebraMap ℝ 𝕜) 1 = 1

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ (algebraMap ℝ 𝕜) 1 = 1
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (x y : ℝ), (algebraMap ℝ 𝕜) (x * y) = (algebraMap ℝ 𝕜) x * (algebraMap ℝ 𝕜) y

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (x y : ℝ), (algebraMap ℝ 𝕜) (x * y) = (algebraMap ℝ 𝕜) x * (algebraMap ℝ 𝕜) y
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ (algebraMap ℝ 𝕜) 0 = 0

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ (algebraMap ℝ 𝕜) 0 = 0
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (x y : ℝ), (algebraMap ℝ 𝕜) (x + y) = (algebraMap ℝ 𝕜) x + (algebraMap ℝ 𝕜) y

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (x y : ℝ), (algebraMap ℝ 𝕜) (x + y) = (algebraMap ℝ 𝕜) x + (algebraMap ℝ 𝕜) y
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ) (x : 𝕜),
    { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r * x =
      x * { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ) (x : 𝕜),
    { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r * x =
      x * { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ) (x : 𝕜),
    r • x = { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r * x

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ) (x : 𝕜),
    r • x = { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r * x
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ) (x : 𝕜), ‖r • x‖ ≤ ‖r‖ * ‖x‖

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ) (x : 𝕜), ‖r • x‖ ≤ ‖r‖ * ‖x‖
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ {f : Filter 𝕜}, Cauchy f → ∃ x, f ≤ nhds x

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ {f : Filter 𝕜}, Cauchy f → ∃ x, f ≤ nhds x
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ 𝕜 →+ ℝ

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ 𝕜 →+ ℝ
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ (⋯ ▸
        let __spread.0 := h.toNormedField;
        re)
      I =
    0

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ (⋯ ▸
        let __spread.0 := h.toNormedField;
        re)
      I =
    0
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ I = 0 ∨ I * I = -1

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ I = 0 ∨ I * I = -1
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
          ((⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z) +
        { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
            ((⋯ ▸
                let __spread.0 := h.toNormedField;
                im)
              z) *
          I =
      z

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
          ((⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z) +
        { toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
            ((⋯ ▸
                let __spread.0 := h.toNormedField;
                im)
              z) *
          I =
      z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          re)
        ({ toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r) =
      r

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          re)
        ({ toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r) =
      r
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          im)
        ({ toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r) =
      0

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (r : ℝ),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          im)
        ({ toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r) =
      0
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z w : 𝕜),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          re)
        (z * w) =
      (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            w -
        (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            w

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z w : 𝕜),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          re)
        (z * w) =
      (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            w -
        (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            w
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z w : 𝕜),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          im)
        (z * w) =
      (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            w +
        (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            w

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z w : 𝕜),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          im)
        (z * w) =
      (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            w +
        (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            w
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          re)
        ((starRingEnd 𝕜) z) =
      (⋯ ▸
          let __spread.0 := h.toNormedField;
          re)
        z

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          re)
        ((starRingEnd 𝕜) z) =
      (⋯ ▸
          let __spread.0 := h.toNormedField;
          re)
        z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          im)
        ((starRingEnd 𝕜) z) =
      -(⋯ ▸
            let __spread.0 := h.toNormedField;
            im)
          z

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    (⋯ ▸
          let __spread.0 := h.toNormedField;
          im)
        ((starRingEnd 𝕜) z) =
      -(⋯ ▸
            let __spread.0 := h.toNormedField;
            im)
          z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ (starRingEnd 𝕜) I = -I

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ (starRingEnd 𝕜) I = -I
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    ‖z‖ ^ 2 =
      (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z +
        (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    ‖z‖ ^ 2 =
      (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z +
        (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z *
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    (⋯ ▸
            let __spread.0 := h.toNormedField;
            im)
          z *
        (⋯ ▸
            let __spread.0 := h.toNormedField;
            im)
          I =
      (⋯ ▸
          let __spread.0 := h.toNormedField;
          im)
        z

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (z : 𝕜),
    (⋯ ▸
            let __spread.0 := h.toNormedField;
            im)
          z *
        (⋯ ▸
            let __spread.0 := h.toNormedField;
            im)
          I =
      (⋯ ▸
          let __spread.0 := h.toNormedField;
          im)
        z
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ {z w : 𝕜},
    z ≤ w ↔
      (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z ≤
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            w ∧
        (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z =
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            w

Tactic:
;

Goal state after:
K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ {z w : 𝕜},
    z ≤ w ↔
      (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            z ≤
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              re)
            w ∧
        (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            z =
          (⋯ ▸
              let __spread.0 := h.toNormedField;
              im)
            w
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: 𝕜 : Type u_3
W : Type u_5
G : Type u_6
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup W
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
e : G ≃ₗᵢ[𝕜] W
α : ↥(unitary 𝕜)
x✝ : W
⊢ (α • e).symm x✝ = (α⁻¹ • e.symm) x✝

Tactic:
;

Goal state after:
𝕜 : Type u_3
W : Type u_5
G : Type u_6
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup W
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
e : G ≃ₗᵢ[𝕜] W
α : ↥(unitary 𝕜)
x✝ : W
⊢ (α • e).symm x✝ = (α⁻¹ • e.symm) x✝
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: 𝕜 : Type u_3
V : Type u_4
W : Type u_5
G : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : Module 𝕜 V
inst✝³ : SeminormedAddCommGroup W
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
α : ↥(unitary 𝕜)
e : V ≃ₗᵢ[𝕜] G
f : G ≃ₗᵢ[𝕜] W
x✝ : V
⊢ ((α • e).trans f) x✝ = (α • e.trans f) x✝

Tactic:
;

Goal state after:
𝕜 : Type u_3
V : Type u_4
W : Type u_5
G : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : Module 𝕜 V
inst✝³ : SeminormedAddCommGroup W
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
α : ↥(unitary 𝕜)
e : V ≃ₗᵢ[𝕜] G
f : G ≃ₗᵢ[𝕜] W
x✝ : V
⊢ ((α • e).trans f) x✝ = (α • e.trans f) x✝
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: 𝕜 : Type u_3
V : Type u_4
W : Type u_5
G : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : Module 𝕜 V
inst✝³ : SeminormedAddCommGroup W
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
α : ↥(unitary 𝕜)
e : V ≃ₗᵢ[𝕜] G
f : G ≃ₗᵢ[𝕜] W
x✝ : V
⊢ (e.trans (α • f)) x✝ = (α • e.trans f) x✝

Tactic:
;

Goal state after:
𝕜 : Type u_3
V : Type u_4
W : Type u_5
G : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : Module 𝕜 V
inst✝³ : SeminormedAddCommGroup W
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
α : ↥(unitary 𝕜)
e : V ≃ₗᵢ[𝕜] G
f : G ≃ₗᵢ[𝕜] W
x✝ : V
⊢ (e.trans (α • f)) x✝ = (α • e.trans f) x✝
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
c : ℝ
f : α → E
g : α → F
l : Filter α
h : IsBigOWith c l f g
⊢ ∃ c, IsBigOWith c l f g

Tactic:
;

Goal state after:
α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
c : ℝ
f : α → E
g : α → F
l : Filter α
h : IsBigOWith c l f g
⊢ ∃ c, IsBigOWith c l f g
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
c : ℝ
f : α → E
g : α → F
l : Filter α
h : IsBigOWith c l f g
⊢ f =O[l] g

Tactic:
rw [IsBigO_def]; exact ⟨c, h⟩

Goal state after: