mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.IsLocalHomeomorph

Goal state before:
pp: case refine_2.refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x : X
U : Set X
hx : x ∈ U
hU : IsOpen[inst✝¹] U
f : OpenPartialHomeomorph X Y
hxf : x ∈ f.source
hf : IsLocalHomeomorph ↑f
⊢ f.toPartialEquiv.2 '' (f.target ∩ ↑f.symm ⁻¹' U) = f.source ∩ U

Tactic:
·

Goal state after:
case refine_2.refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x : X
U : Set X
hx : x ∈ U
hU : IsOpen[inst✝¹] U
f : OpenPartialHomeomorph X Y
hxf : x ∈ f.source
hf : IsLocalHomeomorph ↑f
⊢ f.toPartialEquiv.2 '' (f.target ∩ ↑f.symm ⁻¹' U) = f.source ∩ U
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ ∀
    a_1 ∈
      (fun a_2 ↦ sSup ((fun x ↦ ‖a.toProd.2 (star (a.toProd.1 (star a_2))) * x‖₊) '' Metric.closedBall 0 1)) ''
        Metric.closedBall 0 1,
    a_1 ≤ ‖a‖₊ * ‖a‖₊

pp: case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
⊢ ∃
    a_1 ∈
      (fun a_2 ↦ sSup ((fun x ↦ ‖a.toProd.2 (star (a.toProd.1 (star a_2))) * x‖₊) '' Metric.closedBall 0 1)) ''
        Metric.closedBall 0 1,
    r < a_1

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ ∀
    a_1 ∈
      (fun a_2 ↦ sSup ((fun x ↦ ‖a.toProd.2 (star (a.toProd.1 (star a_2))) * x‖₊) '' Metric.closedBall 0 1)) ''
        Metric.closedBall 0 1,
    a_1 ≤ ‖a‖₊ * ‖a‖₊
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
⊢ ∃
    a_1 ∈
      (fun a_2 ↦ sSup ((fun x ↦ ‖a.toProd.2 (star (a.toProd.1 (star a_2))) * x‖₊) '' Metric.closedBall 0 1)) ''
        Metric.closedBall 0 1,
    r < a_1

Tactic:
·

Goal state after:
case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
⊢ ∃
    a_1 ∈
      (fun a_2 ↦ sSup ((fun x ↦ ‖a.toProd.2 (star (a.toProd.1 (star a_2))) * x‖₊) '' Metric.closedBall 0 1)) ''
        Metric.closedBall 0 1,
    r < a_1
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G E
Y : Type u_4
inst✝ : TopologicalSpace Y
φ : X ≃ₜ Y
h : IsQuotientCoveringMap (⇑φ ∘ f) G
⊢ IsQuotientCoveringMap f G

Tactic:
convert! h.homeomorph_comp φ.symm

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G E
Y : Type u_4
inst✝ : TopologicalSpace Y
φ : X ≃ₜ Y
h : IsQuotientCoveringMap (⇑φ ∘ f) G
⊢ f = ⇑φ.symm ∘ ⇑φ ∘ f
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
G : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G E
hf : IsQuotientCoveringMap f G
x : X
e : ↑(f ⁻¹' {x})
this✝ : MulAction G ↑(f ⁻¹' {x}) := hf.mulActionFiber x
this : MulAction.IsPretransitive G ↑(f ⁻¹' {x})
g : G
⊢ ∃ g_1, ((hf.toPermFiber x) g_1) e = g • e

Tactic:
use g

Goal state after:
case h
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
G : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G E
hf : IsQuotientCoveringMap f G
x : X
e : ↑(f ⁻¹' {x})
this✝ : MulAction G ↑(f ⁻¹' {x}) := ⋯
this : MulAction.IsPretransitive G ↑(f ⁻¹' {x})
g : G
⊢ ((hf.toPermFiber x) g) e = g • e
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: case refine_1
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
e : E
x✝ : e ∈ f ⁻¹' f '' U
e' : E
heU : e' ∈ U
he : f e' = f e
⊢ e ∈ ⋃ g, (fun x ↦ g • x) ⁻¹' U

pp: case refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
e : E
⊢ e ∈ ⋃ g, (fun x ↦ g • x) ⁻¹' U → e ∈ f ⁻¹' f '' U

Tactic:
·

Goal state after:
case refine_1
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
e : E
x✝ : e ∈ f ⁻¹' f '' U
e' : E
heU : e' ∈ U
he : f e' = f e
⊢ e ∈ ⋃ g, (fun x ↦ g • x) ⁻¹' U
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: case refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
e : E
⊢ e ∈ ⋃ g, (fun x ↦ g • x) ⁻¹' U → e ∈ f ⁻¹' f '' U

Tactic:
·

Goal state after:
case refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
e : E
⊢ e ∈ ⋃ g, (fun x ↦ g • x) ⁻¹' U → e ∈ f ⁻¹' f '' U
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: case refine_2.refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
hr' : sqrt r < sSup ((fun x ↦ ‖a.toProd.1 x‖₊) '' Metric.closedBall 0 1)
x : A
hx : x ∈ Metric.closedBall 0 1
hxr : sqrt r < (fun x ↦ ‖a.toProd.1 x‖₊) x
hx' : ‖x‖₊ ≤ 1
⊢ BddAbove ((fun x_1 ↦ ‖a.toProd.2 (star (a.toProd.1 (star (star x)))) * x_1‖₊) '' Metric.closedBall 0 1)

pp: case refine_2.refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
hr' : sqrt r < sSup ((fun x ↦ ‖a.toProd.1 x‖₊) '' Metric.closedBall 0 1)
x : A
hx : x ∈ Metric.closedBall 0 1
hxr : sqrt r < (fun x ↦ ‖a.toProd.1 x‖₊) x
hx' : ‖x‖₊ ≤ 1
⊢ r < (fun x_1 ↦ ‖a.toProd.2 (star (a.toProd.1 (star (star x)))) * x_1‖₊) x

Tactic:
·

Goal state after:
case refine_2.refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
hr' : sqrt r < sSup ((fun x ↦ ‖a.toProd.1 x‖₊) '' Metric.closedBall 0 1)
x : A
hx : x ∈ Metric.closedBall 0 1
hxr : sqrt r < (fun x ↦ ‖a.toProd.1 x‖₊) x
hx' : ‖x‖₊ ≤ 1
⊢ BddAbove ((fun x_1 ↦ ‖a.toProd.2 (star (a.toProd.1 (star (star x)))) * x_1‖₊) '' Metric.closedBall 0 1)
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: case refine_1
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
hW : e ∈ f ⁻¹' W
⊢ e ∈ ⋃ i, (fun x ↦ i • x) ⁻¹' (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)

pp: case refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
⊢ e ∈ ⋃ i, (fun x ↦ i • x) ⁻¹' (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U) → e ∈ f ⁻¹' W

Tactic:
·

Goal state after:
case refine_1
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
hW : e ∈ f ⁻¹' W
⊢ e ∈ ⋃ i, (fun x ↦ i • x) ⁻¹' (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: case refine_1.refine_1
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
hW : e ∈ f ⁻¹' W
g' : G
he' : (fun m ↦ m • e) g' ∈ U
hfe : f ((fun m ↦ m • e) g') = f e
⊢ (fun x ↦ (g⁻¹ * g') • x) e ∈ f ⁻¹' W

pp: case refine_1.refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
hW : e ∈ f ⁻¹' W
g' : G
he' : (fun m ↦ m • e) g' ∈ U
hfe : f ((fun m ↦ m • e) g') = f e
⊢ (fun x ↦ (g⁻¹ * g') • x) e ∈ (fun x ↦ g • x) ⁻¹' U

Tactic:
·

Goal state after:
case refine_1.refine_1
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
hW : e ∈ f ⁻¹' W
g' : G
he' : (fun m ↦ m • e) g' ∈ U
hfe : f ((fun m ↦ m • e) g') = f e
⊢ (fun x ↦ (g⁻¹ * g') • x) e ∈ f ⁻¹' W
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: case refine_1.refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
hW : e ∈ f ⁻¹' W
g' : G
he' : (fun m ↦ m • e) g' ∈ U
hfe : f ((fun m ↦ m • e) g') = f e
⊢ (fun x ↦ (g⁻¹ * g') • x) e ∈ (fun x ↦ g • x) ⁻¹' U

Tactic:
·

Goal state after:
case refine_1.refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
hW : e ∈ f ⁻¹' W
g' : G
he' : (fun m ↦ m • e) g' ∈ U
hfe : f ((fun m ↦ m • e) g') = f e
⊢ (fun x ↦ (g⁻¹ * g') • x) e ∈ (fun x ↦ g • x) ⁻¹' U
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: case refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
⊢ e ∈ ⋃ i, (fun x ↦ i • x) ⁻¹' (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U) → e ∈ f ⁻¹' W

Tactic:
·

Goal state after:
case refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
⊢ e ∈ ⋃ i, (fun x ↦ i • x) ⁻¹' (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U) → e ∈ f ⁻¹' W
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: case refine_2.refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
hr' : sqrt r < sSup ((fun x ↦ ‖a.toProd.1 x‖₊) '' Metric.closedBall 0 1)
x : A
hx : x ∈ Metric.closedBall 0 1
hxr : sqrt r < (fun x ↦ ‖a.toProd.1 x‖₊) x
hx' : ‖x‖₊ ≤ 1
⊢ r < (fun x_1 ↦ ‖a.toProd.2 (star (a.toProd.1 (star (star x)))) * x_1‖₊) x

Tactic:
·

Goal state after:
case refine_2.refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
hr' : sqrt r < sSup ((fun x ↦ ‖a.toProd.1 x‖₊) '' Metric.closedBall 0 1)
x : A
hx : x ∈ Metric.closedBall 0 1
hxr : sqrt r < (fun x ↦ ‖a.toProd.1 x‖₊) x
hx' : ‖x‖₊ ≤ 1
⊢ r < (fun x_1 ↦ ‖a.toProd.2 (star (a.toProd.1 (star (star x)))) * x_1‖₊) x
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: case refine_3
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
e₁ : E
h₁ : e₁ ∈ (fun x ↦ g • x) ⁻¹' U
e₂ : E
h₂ : e₂ ∈ (fun x ↦ g • x) ⁻¹' U
he : f e₁ = f e₂
⊢ e₁ = e₂

pp: case refine_4
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
⊢ ∀ (i : G), Set.SurjOn f ((fun x ↦ i • x) ⁻¹' U) (f '' U)

pp: case refine_5
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g₁ g₂ : G
hne : g₁ ≠ g₂
e : E
x✝ : e ∈ (fun g ↦ (fun x ↦ g • x) ⁻¹' U) g₁ ⊓ (fun g ↦ (fun x ↦ g • x) ⁻¹' U) g₂
h₁ : e ∈ (fun g ↦ (fun x ↦ g • x) ⁻¹' U) g₁
h₂ : e ∈ (fun g ↦ (fun x ↦ g • x) ⁻¹' U) g₂
⊢ (fun x ↦ (g₁ * g₂⁻¹) • x) ((fun x ↦ g₂ • x) e) = (fun x ↦ g₁ • x) e

Tactic:
·

Goal state after:
case refine_3
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
e₁ : E
h₁ : e₁ ∈ (fun x ↦ g • x) ⁻¹' U
e₂ : E
h₂ : e₂ ∈ (fun x ↦ g • x) ⁻¹' U
he : f e₁ = f e₂
⊢ e₁ = e₂
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
I : Type u_3
inst✝¹ : TopologicalSpace I
x : X
h : IsEvenlyCovered f x I
Y : Type u_4
inst✝ : TopologicalSpace Y
g : X ≃ₜ Y
inst : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝²] U
hfU : IsOpen[inst✝³] (f ⁻¹' U)
H : ↑(f ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x
⊢ IsOpen[inst✝³] (⇑g ∘ f ⁻¹' ⇑g '' U)

Tactic:
by simpa [Set.preimage_comp]

Goal state after: