File: Mathlib/Topology/Algebra/Star.lean

full_name: ContinuousStar
code: /-- Basic hypothesis to talk about a topological space with a continuous `star` operator. -/
class ContinuousStar (R : Type*) [TopologicalSpace R] [Star R] : Prop where
  /-- The `star` operator is continuous. -/
  continuous_star : Continuous (star : R → R)
goal: Prop

full_name: ContinuousStar.continuous_star
code: /-- Basic hypothesis to talk about a topological space with a continuous `star` operator. -/
class ContinuousStar (R : Type*) [TopologicalSpace R] [Star R] : Prop where
  /-- The `star` operator is continuous. -/
  continuous_star : Continuous (star : R → R)
goal: Continuous star

full_name: continuousOn_star
code: theorem continuousOn_star {s : Set R} : ContinuousOn star s 
goal: ContinuousOn star s

full_name: continuousWithinAt_star
code: theorem continuousWithinAt_star {s : Set R} {x : R} : ContinuousWithinAt star s x 
goal: ContinuousWithinAt star s x

full_name: continuousAt_star
code: theorem continuousAt_star {x : R} : ContinuousAt star x 
goal: ContinuousAt star x

full_name: tendsto_star
code: theorem tendsto_star (a : R) : Tendsto star (𝓝 a) (𝓝 (star a)) 
goal: Filter.Tendsto star (𝓝 a) (𝓝 (star a))

full_name: Filter.Tendsto.star
code: theorem Filter.Tendsto.star {f : α → R} {l : Filter α} {y : R} (h : Tendsto f l (𝓝 y)) :
    Tendsto (fun x => star (f x)) l (𝓝 (star y)) 
goal: Filter.Tendsto (fun x => star (f x)) l (𝓝 (star y))

full_name: Continuous.star
code: @[continuity, fun_prop]
theorem Continuous.star (hf : Continuous f) : Continuous fun x => star (f x) 
goal: Continuous fun x => star (f x)

full_name: ContinuousAt.star
code: @[fun_prop]
theorem ContinuousAt.star (hf : ContinuousAt f x) : ContinuousAt (fun x => star (f x)) x 
goal: ContinuousAt (fun x => star (f x)) x

full_name: ContinuousOn.star
code: @[fun_prop]
theorem ContinuousOn.star (hf : ContinuousOn f s) : ContinuousOn (fun x => star (f x)) s 
goal: ContinuousOn (fun x => star (f x)) s

full_name: ContinuousWithinAt.star
code: theorem ContinuousWithinAt.star (hf : ContinuousWithinAt f s x) :
    ContinuousWithinAt (fun x => star (f x)) s x 
goal: ContinuousWithinAt (fun x => star (f x)) s x

full_name: starContinuousMap
code: /-- The star operation bundled as a continuous map. -/
@[simps]
def starContinuousMap : C(R, R) 
goal: C(R, R)

full_name: starContinuousMap_apply
code: /-- The star operation bundled as a continuous map. -/
@[simps]
def starContinuousMap : C(R, R) 
goal: ∀ (a : R), starContinuousMap a = star a

full_name: instContinuousStarProd
code: instance [Star R] [Star S] [TopologicalSpace R] [TopologicalSpace S] [ContinuousStar R]
    [ContinuousStar S] : ContinuousStar (R × S) 
goal: ContinuousStar (R × S)

full_name: instContinuousStarForall
code: instance {C : ι → Type*} [∀ i, TopologicalSpace (C i)] [∀ i, Star (C i)]
    [∀ i, ContinuousStar (C i)] : ContinuousStar (∀ i, C i) 
goal: ContinuousStar ((i : ι) → C i)

full_name: instContinuousStarMulOpposite
code: instance [Star R] [TopologicalSpace R] [ContinuousStar R] : ContinuousStar Rᵐᵒᵖ 
goal: ContinuousStar Rᵐᵒᵖ

full_name: instContinuousStarUnits
code: instance [Monoid R] [StarMul R] [TopologicalSpace R] [ContinuousStar R] :
    ContinuousStar Rˣ 
goal: ContinuousStar Rˣ