File: Mathlib/Topology/Algebra/Field.lean

full_name: Filter.tendsto_cocompact_mul_left₀
code: /-- Left-multiplication by a nonzero element of a topological division ring is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_left₀ [ContinuousMul K] {a : K} (ha : a ≠ 0) :
    Filter.Tendsto (fun x : K => a * x) (Filter.cocompact K) (Filter.cocompact K) 
goal: Filter.Tendsto (fun x => a * x) (Filter.cocompact K) (Filter.cocompact K)

full_name: Filter.tendsto_cocompact_mul_right₀
code: /-- Right-multiplication by a nonzero element of a topological division ring is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_right₀ [ContinuousMul K] {a : K} (ha : a ≠ 0) :
    Filter.Tendsto (fun x : K => x * a) (Filter.cocompact K) (Filter.cocompact K) 
goal: Filter.Tendsto (fun x => x * a) (Filter.cocompact K) (Filter.cocompact K)

full_name: TopologicalDivisionRing
code: /-- A topological division ring is a division ring with a topology where all operations are
    continuous, including inversion. -/
class TopologicalDivisionRing extends TopologicalRing K, HasContinuousInv₀ K : Prop
goal: Prop

full_name: Subfield.topologicalClosure
code: /-- The (topological-space) closure of a subfield of a topological field is
itself a subfield. -/
def Subfield.topologicalClosure (K : Subfield α) : Subfield α 
goal: Subfield α

full_name: Subfield.le_topologicalClosure
code: theorem Subfield.le_topologicalClosure (s : Subfield α) : s ≤ s.topologicalClosure 
goal: s ≤ s.topologicalClosure

full_name: Subfield.isClosed_topologicalClosure
code: theorem Subfield.isClosed_topologicalClosure (s : Subfield α) :
    IsClosed (s.topologicalClosure : Set α) 
goal: IsClosed ↑s.topologicalClosure

full_name: Subfield.topologicalClosure_minimal
code: theorem Subfield.topologicalClosure_minimal (s : Subfield α) {t : Subfield α} (h : s ≤ t)
    (ht : IsClosed (t : Set α)) : s.topologicalClosure ≤ t 
goal: s.topologicalClosure ≤ t

full_name: affineHomeomorph_symm_apply
code: /--
The map `fun x => a * x + b`, as a homeomorphism from `𝕜` (a topological field) to itself,
when `a ≠ 0`.
-/
@[simps]
def affineHomeomorph (a b : 𝕜) (h : a ≠ 0) : 𝕜 ≃ₜ 𝕜 
goal: (affineHomeomorph a b h).symm y = (y - b) / a

full_name: affineHomeomorph_apply
code: /--
The map `fun x => a * x + b`, as a homeomorphism from `𝕜` (a topological field) to itself,
when `a ≠ 0`.
-/
@[simps]
def affineHomeomorph (a b : 𝕜) (h : a ≠ 0) : 𝕜 ≃ₜ 𝕜 
goal: (affineHomeomorph a b h) x = a * x + b

full_name: affineHomeomorph
code: /--
The map `fun x => a * x + b`, as a homeomorphism from `𝕜` (a topological field) to itself,
when `a ≠ 0`.
-/
@[simps]
def affineHomeomorph (a b : 𝕜) (h : a ≠ 0) : 𝕜 ≃ₜ 𝕜 
goal: 𝕜 ≃ₜ 𝕜

full_name: IsLocalMin.inv
code: theorem IsLocalMin.inv {f : α → β} {a : α} (h1 : IsLocalMin f a) (h2 : ∀ᶠ z in 𝓝 a, 0 < f z) :
    IsLocalMax f⁻¹ a 
goal: IsLocalMax f⁻¹ a

full_name: IsPreconnected.eq_one_or_eq_neg_one_of_sq_eq
code: /-- If `f` is a function `α → 𝕜` which is continuous on a preconnected set `S`, and
`f ^ 2 = 1` on `S`, then either `f = 1` on `S`, or `f = -1` on `S`. -/
theorem IsPreconnected.eq_one_or_eq_neg_one_of_sq_eq [Ring 𝕜] [NoZeroDivisors 𝕜]
    (hS : IsPreconnected S) (hf : ContinuousOn f S) (hsq : EqOn (f ^ 2) 1 S) :
    EqOn f 1 S ∨ EqOn f (-1) S 
goal: Set.EqOn f 1 S ∨ Set.EqOn f (-1) S

full_name: IsPreconnected.eq_or_eq_neg_of_sq_eq
code: /-- If `f, g` are functions `α → 𝕜`, both continuous on a preconnected set `S`, with
`f ^ 2 = g ^ 2` on `S`, and `g z ≠ 0` all `z ∈ S`, then either `f = g` or `f = -g` on
`S`. -/
theorem IsPreconnected.eq_or_eq_neg_of_sq_eq [Field 𝕜] [HasContinuousInv₀ 𝕜] [ContinuousMul 𝕜]
    (hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S)
    (hsq : EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) :
    EqOn f g S ∨ EqOn f (-g) S 
goal: Set.EqOn f g S ∨ Set.EqOn f (-g) S

full_name: IsPreconnected.eq_of_sq_eq
code: /-- If `f, g` are functions `α → 𝕜`, both continuous on a preconnected set `S`, with
`f ^ 2 = g ^ 2` on `S`, and `g z ≠ 0` all `z ∈ S`, then as soon as `f = g` holds at
one point of `S` it holds for all points. -/
theorem IsPreconnected.eq_of_sq_eq [Field 𝕜] [HasContinuousInv₀ 𝕜] [ContinuousMul 𝕜]
    (hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S)
    (hsq : EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) {y : α} (hy : y ∈ S)
    (hy' : f y = g y) : EqOn f g S 
goal: Set.EqOn f g S