File: Mathlib/Analysis/Normed/Ring/Units.lean

full_name: Units.val_oneSub
code: /-- In a complete normed ring, a perturbation of `1` by an element `t` of distance less than `1`
from `1` is a unit.  Here we construct its `Units` structure.  -/
@[simps val]
def oneSub (t : R) (h : ‖t‖ < 1) : Rˣ 
goal: ↑(Units.oneSub t h) = 1 - t

full_name: Units.oneSub
code: /-- In a complete normed ring, a perturbation of `1` by an element `t` of distance less than `1`
from `1` is a unit.  Here we construct its `Units` structure.  -/
@[simps val]
def oneSub (t : R) (h : ‖t‖ < 1) : Rˣ 
goal: Rˣ

full_name: Units.val_add
code: /-- In a complete normed ring, a perturbation of a unit `x` by an element `t` of distance less than
`‖x⁻¹‖⁻¹` from `x` is a unit.  Here we construct its `Units` structure. -/
@[simps! val]
def add (x : Rˣ) (t : R) (h : ‖t‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ 
goal: ↑(x.add t h) = ↑x + t

full_name: Units.add
code: /-- In a complete normed ring, a perturbation of a unit `x` by an element `t` of distance less than
`‖x⁻¹‖⁻¹` from `x` is a unit.  Here we construct its `Units` structure. -/
@[simps! val]
def add (x : Rˣ) (t : R) (h : ‖t‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ 
goal: Rˣ

full_name: Units.val_ofNearby
code: /-- In a complete normed ring, an element `y` of distance less than `‖x⁻¹‖⁻¹` from `x` is a unit.
Here we construct its `Units` structure. -/
@[simps! val]
def ofNearby (x : Rˣ) (y : R) (h : ‖y - x‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ 
goal: ↑(x.ofNearby y h) = y

full_name: Units.ofNearby
code: /-- In a complete normed ring, an element `y` of distance less than `‖x⁻¹‖⁻¹` from `x` is a unit.
Here we construct its `Units` structure. -/
@[simps! val]
def ofNearby (x : Rˣ) (y : R) (h : ‖y - x‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ 
goal: Rˣ

full_name: Units.isOpen
code: /-- The group of units of a complete normed ring is an open subset of the ring. -/
protected theorem isOpen : IsOpen { x : R | IsUnit x } 
goal: IsOpen {x | IsUnit x}

full_name: Units.nhds
code: protected theorem nhds (x : Rˣ) : { x : R | IsUnit x } ∈ 𝓝 (x : R) 
goal: {x | IsUnit x} ∈ 𝓝 ↑x

full_name: nonunits.subset_compl_ball
code: /-- The `nonunits` in a complete normed ring are contained in the complement of the ball of radius
`1` centered at `1 : R`. -/
theorem subset_compl_ball : nonunits R ⊆ (Metric.ball (1 : R) 1)ᶜ 
goal: nonunits R ⊆ (Metric.ball 1 1)ᶜ

full_name: nonunits.isClosed
code: protected theorem isClosed : IsClosed (nonunits R) 
goal: IsClosed (nonunits R)

full_name: NormedRing.inverse_one_sub
code: theorem inverse_one_sub (t : R) (h : ‖t‖ < 1) : inverse (1 - t) = ↑(Units.oneSub t h)⁻¹ 
goal: Ring.inverse (1 - t) = ↑(Units.oneSub t h)⁻¹

full_name: NormedRing.inverse_add
code: /-- The formula `Ring.inverse (x + t) = Ring.inverse (1 + x⁻¹ * t) * x⁻¹` holds for `t` sufficiently
small. -/
theorem inverse_add (x : Rˣ) :
    ∀ᶠ t in 𝓝 0, inverse ((x : R) + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹ 
goal: ∀ᶠ (t : R) in 𝓝 0, Ring.inverse (↑x + t) = Ring.inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹

full_name: NormedRing.inverse_one_sub_nth_order'
code: theorem inverse_one_sub_nth_order' (n : ℕ) {t : R} (ht : ‖t‖ < 1) :
    inverse ((1 : R) - t) = (∑ i ∈ range n, t ^ i) + t ^ n * inverse (1 - t) 
goal: Ring.inverse (1 - t) = ∑ i ∈ Finset.range n, t ^ i + t ^ n * Ring.inverse (1 - t)

full_name: NormedRing.inverse_one_sub_nth_order
code: theorem inverse_one_sub_nth_order (n : ℕ) :
    ∀ᶠ t in 𝓝 0, inverse ((1 : R) - t) = (∑ i ∈ range n, t ^ i) + t ^ n * inverse (1 - t) 
goal: ∀ᶠ (t : R) in 𝓝 0, Ring.inverse (1 - t) = ∑ i ∈ Finset.range n, t ^ i + t ^ n * Ring.inverse (1 - t)

full_name: NormedRing.inverse_add_nth_order
code: /-- The formula
`Ring.inverse (x + t) =
  (∑ i ∈ Finset.range n, (- x⁻¹ * t) ^ i) * x⁻¹ + (- x⁻¹ * t) ^ n * Ring.inverse (x + t)`
holds for `t` sufficiently small. -/
theorem inverse_add_nth_order (x : Rˣ) (n : ℕ) :
    ∀ᶠ t in 𝓝 0, inverse ((x : R) + t) =
      (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * t) ^ n * inverse (x + t) 
goal: ∀ᶠ (t : R) in 𝓝 0,
  Ring.inverse (↑x + t) = (∑ i ∈ Finset.range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * t) ^ n * Ring.inverse (↑x + t)

full_name: NormedRing.inverse_one_sub_norm
code: theorem inverse_one_sub_norm : (fun t : R => inverse (1 - t)) =O[𝓝 0] (fun _t => 1 : R → ℝ) 
goal: (fun t => Ring.inverse (1 - t)) =O[𝓝 0] fun _t => 1

full_name: NormedRing.inverse_add_norm
code: /-- The function `fun t ↦ inverse (x + t)` is O(1) as `t → 0`. -/
theorem inverse_add_norm (x : Rˣ) : (fun t : R => inverse (↑x + t)) =O[𝓝 0] fun _t => (1 : ℝ) 
goal: (fun t => Ring.inverse (↑x + t)) =O[𝓝 0] fun _t => 1

full_name: NormedRing.inverse_add_norm_diff_nth_order
code: /-- The function
`fun t ↦ Ring.inverse (x + t) - (∑ i ∈ Finset.range n, (- x⁻¹ * t) ^ i) * x⁻¹`
is `O(t ^ n)` as `t → 0`. -/
theorem inverse_add_norm_diff_nth_order (x : Rˣ) (n : ℕ) :
    (fun t : R => inverse (↑x + t) - (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 (0 : R)]
      fun t => ‖t‖ ^ n 
goal: (fun t => Ring.inverse (↑x + t) - (∑ i ∈ Finset.range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ n

full_name: NormedRing.inverse_add_norm_diff_first_order
code: /-- The function `fun t ↦ Ring.inverse (x + t) - x⁻¹` is `O(t)` as `t → 0`. -/
theorem inverse_add_norm_diff_first_order (x : Rˣ) :
    (fun t : R => inverse (↑x + t) - ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ 
goal: (fun t => Ring.inverse (↑x + t) - ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖

full_name: NormedRing.inverse_add_norm_diff_second_order
code: /-- The function `fun t ↦ Ring.inverse (x + t) - x⁻¹ + x⁻¹ * t * x⁻¹` is `O(t ^ 2)` as `t → 0`. -/
theorem inverse_add_norm_diff_second_order (x : Rˣ) :
    (fun t : R => inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ 2 
goal: (fun t => Ring.inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ 2

full_name: NormedRing.inverse_continuousAt
code: /-- The function `Ring.inverse` is continuous at each unit of `R`. -/
theorem inverse_continuousAt (x : Rˣ) : ContinuousAt inverse (x : R) 
goal: ContinuousAt Ring.inverse ↑x

full_name: Units.openEmbedding_val
code: /-- In a normed ring, the coercion from `Rˣ` (equipped with the induced topology from the
embedding in `R × R`) to `R` is an open embedding. -/
theorem openEmbedding_val : OpenEmbedding (val : Rˣ → R) 
goal: OpenEmbedding Units.val

full_name: Units.isOpenMap_val
code: /-- In a normed ring, the coercion from `Rˣ` (equipped with the induced topology from the
embedding in `R × R`) to `R` is an open map. -/
theorem isOpenMap_val : IsOpenMap (val : Rˣ → R) 
goal: IsOpenMap Units.val

full_name: Ideal.eq_top_of_norm_lt_one
code: /-- An ideal which contains an element within `1` of `1 : R` is the unit ideal. -/
theorem eq_top_of_norm_lt_one (I : Ideal R) {x : R} (hxI : x ∈ I) (hx : ‖1 - x‖ < 1) : I = ⊤ 
goal: I = ⊤

full_name: Ideal.closure_ne_top
code: /-- The `Ideal.closure` of a proper ideal in a complete normed ring is proper. -/
theorem closure_ne_top (I : Ideal R) (hI : I ≠ ⊤) : I.closure ≠ ⊤ 
goal: I.closure ≠ ⊤

full_name: Ideal.IsMaximal.closure_eq
code: /-- The `Ideal.closure` of a maximal ideal in a complete normed ring is the ideal itself. -/
theorem IsMaximal.closure_eq {I : Ideal R} (hI : I.IsMaximal) : I.closure = I 
goal: I.closure = I

full_name: Ideal.IsMaximal.isClosed
code: /-- Maximal ideals in complete normed rings are closed. -/
instance IsMaximal.isClosed {I : Ideal R} [hI : I.IsMaximal] : IsClosed (I : Set R) 
goal: IsClosed ↑I