mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
⊢ Tendsto (fun x ↦ f x / r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁸ : UniformSpace α
inst✝⁷ : Group α
inst✝⁶ : IsUniformGroup α
hom : Type u_3
inst✝⁵ : UniformSpace β
inst✝⁴ : DiscreteTopology β
inst✝³ : Group β
inst✝² : IsUniformGroup β
inst✝¹ : FunLike hom α β
inst✝ : MonoidHomClass hom α β
f : hom
⊢ UniformContinuous ⇑f ↔ IsOpen[inst✝⁸.toTopologicalSpace] ↑(↑f).ker

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
⊢ Tendsto (fun x ↦ r * f x) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
⊢ Tendsto (fun x ↦ f x * r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁷ : UniformSpace α
inst✝⁶ : Group α
inst✝⁵ : IsUniformGroup α
hom : Type u_3
inst✝⁴ : UniformSpace β
inst✝³ : Group β
inst✝² : IsUniformGroup β
inst✝¹ : FunLike hom α β
inst✝ : MonoidHomClass hom α β
f : hom
h : Continuous[inst✝⁷.toTopologicalSpace, inst✝⁴.toTopologicalSpace] ⇑f
this : Tendsto (⇑f) (𝓝 1) (𝓝 (f 1))
⊢ Tendsto (⇑f) (𝓝 1) (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁷ : UniformSpace α
inst✝⁶ : Group α
inst✝⁵ : IsUniformGroup α
Hom : Type u_3
inst✝⁴ : UniformSpace β
inst✝³ : Group β
inst✝² : IsUniformGroup β
inst✝¹ : FunLike Hom α β
inst✝ : MonoidHomClass Hom α β
f : Hom
h : IsInducing ⇑f
⊢ comap (fun x ↦ (f x.1, f x.2)) (𝓤 β) = 𝓤 α

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
⊢ Tendsto (fun x ↦ f x / r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atBot
⊢ Tendsto (fun x ↦ r * f x) l atTop ↔ r < 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
this : Tendsto (fun p ↦ (p.2 * p.1⁻¹)⁻¹) (comap (fun p ↦ p.2 * p.1⁻¹) (𝓝 1)) (𝓝 1⁻¹)
⊢ Tendsto Prod.swap (comap (fun p ↦ p.2 * p.1⁻¹) (𝓝 1)) (comap (fun p ↦ p.2 * p.1⁻¹) (𝓝 1))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atBot
⊢ Tendsto (fun x ↦ f x * r) l atTop ↔ r < 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atBot
⊢ Tendsto (fun x ↦ f x / r) l atTop ↔ r < 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atBot
⊢ Tendsto (fun x ↦ r * f x) l atBot ↔ 0 < r

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atBot
⊢ Tendsto (fun x ↦ f x * r) l atBot ↔ 0 < r

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atBot
⊢ Tendsto (fun x ↦ f x / r) l atBot ↔ 0 < r

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
U : Set G
H : U ∈ 𝓝 1
⊢ U ∈ map (fun p ↦ p.2 * p.1⁻¹) ((comap (fun p ↦ p.2 * p.1⁻¹) (𝓝 1)).lift' fun s ↦ SetRel.comp s s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
x✝ : G
⊢ 𝓝 x✝ = comap (Prod.mk x✝) (comap (fun p ↦ p.2 * p.1⁻¹) (𝓝 1))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atTop
⊢ Tendsto (fun x ↦ r * f x) l atBot ↔ r < 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atTop
⊢ Tendsto (fun x ↦ f x * r) l atBot ↔ r < 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
this : Tendsto (fun p ↦ (p.1⁻¹ * p.2)⁻¹) (comap (fun p ↦ p.1⁻¹ * p.2) (𝓝 1)) (𝓝 1⁻¹)
⊢ Tendsto Prod.swap (comap (fun p ↦ p.1⁻¹ * p.2) (𝓝 1)) (comap (fun p ↦ p.1⁻¹ * p.2) (𝓝 1))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atTop
⊢ Tendsto (fun x ↦ f x / r) l atBot ↔ r < 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Field

Goal state before:
pp: α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
c : α
n : ℕ
⊢ Tendsto (fun x ↦ c * x ^ n) atTop atBot ↔ n ≠ 0 ∧ c < 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
K U : Set G
hK : IsCompact K
hU : IsOpen[inst✝²] U
hKU : K ⊆ U
⊢ ∃ V ∈ 𝓝 1, K * V ⊆ U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
U : Set G
H : U ∈ 𝓝 1
⊢ U ∈ map (fun p ↦ p.1⁻¹ * p.2) ((comap (fun p ↦ p.1⁻¹ * p.2) (𝓝 1)).lift' fun s ↦ SetRel.comp s s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
x✝ : G
⊢ 𝓝 x✝ = comap (Prod.mk x✝) (comap (fun p ↦ p.1⁻¹ * p.2) (𝓝 1))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : CommGroup G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
⊢ (fun x ↦ x.1.2 * x.2.2⁻¹ * (x.2.1 * x.1.1⁻¹)) = (fun p ↦ p.1 * p.2⁻¹) ∘ fun p ↦ (p.1.2 * p.1.1⁻¹, p.2.2 * p.2.1⁻¹)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
K U : Set G
hK : IsCompact K
hU : IsOpen[inst✝²] U
hKU : K ⊆ U
⊢ ∃ V ∈ 𝓝 1, V * K ⊆ U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
K V : Set G
hK : IsCompact K
g₀ : G
hg₀ : g₀ ∈ interior V
g : G
x✝ : g ∈ K
⊢ Continuous[inst✝², inst✝²] fun x ↦ g₀ * g⁻¹ * x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
K V : Set G
hK : IsCompact K
hV : (interior V).Nonempty
⊢ ∃ t, K ⊆ ⋃ x ∈ t, interior ((fun x_1 ↦ x * x_1) ⁻¹' V)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
K V : Set G
hK : IsCompact K
hV : (interior V).Nonempty
⊢ ∃ t, K ⊆ ⋃ g ∈ t, (fun x ↦ g * x) ⁻¹' V

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : CommGroup G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
this :
  (fun x ↦ x.1.2 * x.2.2⁻¹ * (x.2.1 * x.1.1⁻¹)) = (fun p ↦ p.1 * p.2⁻¹) ∘ fun p ↦ (p.1.2 * p.1.1⁻¹, p.2.2 * p.2.1⁻¹)
⊢ Continuous[instTopologicalSpaceProd, inst✝¹] fun p ↦ p.1 * p.2⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_1
inst✝² : CommGroup G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
⊢ IsUniformGroup G

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Defs

Goal state before:
pp: G : Type u_2
u : UniformSpace G
inst✝¹ : Group G
inst✝ : IsUniformGroup G
⊢ IsTopologicalGroup.rightUniformSpace G = u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
H : Type x
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace G
inst✝³ : Group G
inst✝² : IsTopologicalGroup G
inst✝¹ : SeparableSpace G
inst✝ : WeaklyLocallyCompactSpace G
L : Set G
hLc : IsCompact L
hL1 : L ∈ 𝓝 1
x : G
⊢ (range (denseSeq G) ∩ (fun y ↦ x * y) ⁻¹' L).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
H : Type x
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace G
inst✝³ : Group G
inst✝² : IsTopologicalGroup G
inst✝¹ : SeparableSpace G
inst✝ : WeaklyLocallyCompactSpace G
⊢ SigmaCompactSpace G

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : NoncompactSpace G
K L : Set G
hK : IsCompact K
hL : IsCompact L
A : ¬K * L⁻¹ = univ
⊢ ∃ g, g ∉ K * L⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : NoncompactSpace G
K L : Set G
hK : IsCompact K
hL : IsCompact L
A : ¬K * L⁻¹ = univ
g : G
hg : g ∉ K * L⁻¹
b : G
bL : b ∈ L
ha : (fun x ↦ g • x) b ∈ K
⊢ b⁻¹ ∈ L⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : NoncompactSpace G
K L : Set G
hK : IsCompact K
hL : IsCompact L
⊢ ∃ g, Disjoint K (g • L)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
x y : G
⊢ 𝓝 (x * y) = map (fun x_1 ↦ x * x_1) (map (fun x ↦ x * y) (𝓝 1 * 𝓝 1))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
x y : G
⊢ map (fun x_1 ↦ x * x_1) (map (fun x ↦ x * y) (𝓝 1 * 𝓝 1)) = map₂ (fun a b ↦ x * (a * b * y)) (𝓝 1) (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
x y : G
⊢ map₂ (fun a b ↦ x * (a * b * y)) (𝓝 1) (𝓝 1) = map₂ (fun a b ↦ x * a * (b * y)) (𝓝 1) (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
x y : G
⊢ map₂ (fun a b ↦ x * a * (b * y)) (𝓝 1) (𝓝 1) = 𝓝 x * 𝓝 y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
H : Type x
inst✝⁴ : Group G
inst✝³ : Monoid H
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace H
inst✝ : ContinuousInv G
f : G →* Hˣ
hf : Continuous[inst✝², inst✝¹] ⇑((Units.coeHom H).comp f)
⊢ Continuous[inst✝², Units.instTopologicalSpaceUnits] ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: α : Type u
inst✝³ : Monoid α
inst✝² : TopologicalSpace α
inst✝¹ : T1Space α
inst✝ : ContinuousMul α
⊢ IsClosed[instTopologicalSpaceProd] (range ⇑(embedProduct α))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝⁵ : Monoid α
inst✝⁴ : TopologicalSpace α
inst✝³ : Monoid β
inst✝² : TopologicalSpace β
inst✝¹ : ContinuousMul α
inst✝ : T1Space α
f : α →* β
hf : IsClosedEmbedding ⇑f
⊢ IsClosedEmbedding ⇑(map f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: α : Type u
inst✝³ : Monoid α
inst✝² : TopologicalSpace α
inst✝¹ : T1Space α
inst✝ : ContinuousMul α
S : Submonoid α
hS : IsCompact ↑S
⊢ IsCompact ↑S.units

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
ι : Sort u_1
inst✝ : Group G
ts' : ι → TopologicalSpace G
h' : ∀ (i : ι), IsTopologicalGroup G
⊢ IsTopologicalGroup G

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Basic

Goal state before:
pp: G : Type w
inst✝ : Group G
t₁ t₂ : TopologicalSpace G
h₁ : IsTopologicalGroup G
h₂ : IsTopologicalGroup G
⊢ IsTopologicalGroup G

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
a : α
⊢ comap (fun x ↦ (x.1 * a, x.2 * a)) (𝓤 α) = 𝓤 α

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u_4
inst✝² : Group G
inst✝¹ : UniformSpace G
inst✝ : IsUniformGroup G
𝓕 : Filter G
⊢ Cauchy 𝓕 ↔ 𝓕.NeBot ∧ Tendsto (fun p ↦ p.1 / p.2) (𝓕 ×ˢ 𝓕) (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u_4
inst✝² : Group G
inst✝¹ : UniformSpace G
inst✝ : IsUniformGroup G
𝓕 : Filter G
⊢ Cauchy 𝓕 ↔ 𝓕.NeBot ∧ Tendsto (fun p ↦ p.2 / p.1) (𝓕 ×ˢ 𝓕) (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: ι : Type u_3
G : Type u_4
inst✝² : Group G
inst✝¹ : UniformSpace G
inst✝ : IsUniformGroup G
𝓕 : Filter ι
f : ι → G
⊢ Cauchy (map f 𝓕) ↔ 𝓕.NeBot ∧ Tendsto (fun p ↦ f p.1 / f p.2) (𝓕 ×ˢ 𝓕) (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: ι : Type u_3
G : Type u_4
inst✝² : Group G
inst✝¹ : UniformSpace G
inst✝ : IsUniformGroup G
𝓕 : Filter ι
f : ι → G
⊢ Cauchy (map f 𝓕) ↔ 𝓕.NeBot ∧ Tendsto (fun p ↦ f p.2 / f p.1) (𝓕 ×ˢ 𝓕) (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u_3
inst✝³ : Group G
inst✝² : UniformSpace G
inst✝¹ : IsRightUniformGroup G
inst✝ : WeaklyLocallyCompactSpace G
f : Filter G
hf : Cauchy f
this : f.NeBot
K : Set G
K_compact : IsCompact K
K_mem : K ∈ 𝓝 1
⊢ ∃ x, ∀ᶠ (y : G) in f, y / x ∈ K

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u_3
inst✝³ : Group G
inst✝² : UniformSpace G
inst✝¹ : IsRightUniformGroup G
inst✝ : WeaklyLocallyCompactSpace G
f : Filter G
hf : Cauchy f
this : f.NeBot
K : Set G
K_compact : IsCompact K
K_mem : K ∈ 𝓝 1
x : G
hx : ∀ᶠ (y : G) in f, y ∈ MulOpposite.op x • K
Kx_complete : IsComplete (MulOpposite.op x • K)
⊢ f ≤ 𝓟 (MulOpposite.op x • K)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
G₀ : Type u_3
inst✝² : DivInvMonoid G₀
inst✝¹ : TopologicalSpace G₀
inst✝ : SeparatelyContinuousMul G₀
f : α → G₀
l : Filter α
x : G₀
hf : Tendsto f l (𝓝 x)
y : G₀
⊢ Tendsto (fun a ↦ f a / y) l (𝓝 (x / y))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u_3
inst✝³ : Group G
inst✝² : UniformSpace G
inst✝¹ : IsRightUniformGroup G
inst✝ : WeaklyLocallyCompactSpace G
f : Filter G
hf : Cauchy f
⊢ ∃ x, f ≤ 𝓝 x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
G₀ : Type u_3
inst✝³ : DivInvMonoid G₀
inst✝² : TopologicalSpace G₀
inst✝¹ : SeparatelyContinuousMul G₀
f : α → G₀
s : Set α
inst✝ : TopologicalSpace α
hf : ContinuousOn f s
y : G₀
⊢ ContinuousOn (fun x ↦ f x / y) s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
G₀ : Type u_3
inst✝³ : DivInvMonoid G₀
inst✝² : TopologicalSpace G₀
inst✝¹ : SeparatelyContinuousMul G₀
f : α → G₀
inst✝ : TopologicalSpace α
hf : Continuous[inst✝, inst✝²] f
y : G₀
⊢ Continuous[inst✝, inst✝²] fun x ↦ f x / y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
β : Type u_2
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : TopologicalSpace G₀
inst✝ : ContinuousInv₀ G₀
x : G₀
⊢ (∃ x_1, x_1 ≠ 0 ∧ x_1 = x) ↔ x ∈ {g | g = 0 → False}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
s : Set α
⊢ (∀ i ∈ 𝓝 1, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ {x | x.2⁻¹ * x.1 ∈ id i}}) ↔
    ∀ U ∈ 𝓝 1, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y • U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
β : Type u_2
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : TopologicalSpace G₀
inst✝ : ContinuousInv₀ G₀
x✝ : { g // g ≠ 0 }
⊢ (fun g ↦ ⟨(↑g)⁻¹, ⋯⟩) ((fun g ↦ ⟨(↑g)⁻¹, ⋯⟩) x✝) = x✝

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
s : Set α
hs : TotallyBounded s
⊢ TotallyBounded s⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
s : Set α
hs : TotallyBounded s⁻¹
⊢ TotallyBounded s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : TopologicalSpace G₀
inst✝ : ContinuousInv₀ G₀
x : G₀
hx : x ≠ 0
⊢ 𝓝 x⁻¹ = (𝓝 x)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : TopologicalSpace G₀
inst✝ : ContinuousInv₀ G₀
x : G₀
l : Filter α
f : α → G₀
hx : x ≠ 0
⊢ Tendsto (fun x ↦ (f x)⁻¹) l (𝓝 x⁻¹) ↔ Tendsto f l (𝓝 x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : TopologicalSpace G₀
inst✝¹ : ContinuousInv₀ G₀
inst✝ : ContinuousMul G₀
f g : α → G₀
l : Filter α
a b : G₀
hf : Tendsto f l (𝓝 a)
hg : Tendsto g l (𝓝 b)
hy : b ≠ 0
⊢ Tendsto (f / g) l (𝓝 (a / b))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
ι : Type u_3
l : Filter ι
f f' : ι → β → α
s : Set β
hf : UniformCauchySeqOn f l s
hf' : UniformCauchySeqOn f' l s
u : Set (α × α)
hu : u ∈ 𝓤 α
⊢ ∀ᶠ (m : ι × ι) in l ×ˢ l, ∀ x ∈ s, ((f * f') m.1 x, (f * f') m.2 x) ∈ u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
ι : Type u_3
l : Filter ι
f f' : ι → β → α
s : Set β
hf : UniformCauchySeqOn f l s
hf' : UniformCauchySeqOn f' l s
u : Set (α × α)
hu : u ∈ 𝓤 α
⊢ ∀ᶠ (m : ι × ι) in l ×ˢ l, ∀ x ∈ s, ((f / f') m.1 x, (f / f') m.2 x) ∈ u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
G₀ : Type u_3
inst✝⁴ : GroupWithZero G₀
inst✝³ : TopologicalSpace G₀
inst✝² : ContinuousInv₀ G₀
inst✝¹ : ContinuousMul G₀
inst✝ : T1Space G₀
f g : α → G₀
l : Filter α
x y : G₀
hg : Tendsto g l (𝓝 y)
hy : y ≠ 0
⊢ Tendsto (fun n ↦ f n * g n) l (𝓝 (x * y)) ↔ Tendsto f l (𝓝 x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
G₀ : Type u_3
inst✝⁴ : GroupWithZero G₀
inst✝³ : TopologicalSpace G₀
inst✝² : ContinuousInv₀ G₀
inst✝¹ : ContinuousMul G₀
f g : α → G₀
inst✝ : TopologicalSpace α
hf : Continuous[inst✝, inst✝³] f
hg : Continuous[inst✝, inst✝³] g
h₀ : ∀ (x : α), g x ≠ 0
⊢ Continuous[inst✝, inst✝³] (f / g)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
ι : Type u_3
l : Filter ι
f : ι → β → α
s : Set β
hf : UniformCauchySeqOn f l s
u : Set (α × α)
hu : u ∈ 𝓤 α
⊢ ∀ᶠ (m : ι × ι) in l ×ˢ l, ∀ x ∈ s, (f⁻¹ m.1 x, f⁻¹ m.2 x) ∈ u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.GroupTopology

Goal state before:
pp: α : Type u
inst✝ : Group α
g : GroupTopology α
⊢ Continuous[instTopologicalSpaceProd, g.toTopologicalSpace] fun p ↦ p.1 * p.2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: α : Type u_1
G₀ : Type u_3
inst✝⁴ : GroupWithZero G₀
inst✝³ : TopologicalSpace G₀
inst✝² : ContinuousInv₀ G₀
inst✝¹ : ContinuousMul G₀
f g : α → G₀
inst✝ : TopologicalSpace α
hf : Continuous[inst✝, inst✝³] f
hg : Continuous[inst✝, inst✝³] g
h₀ : ∀ (x : α), g x ≠ 0
⊢ Continuous[inst✝, inst✝³] fun x ↦ f x / g x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.GroupTopology

Goal state before:
pp: α : Type u
inst✝ : Group α
g : GroupTopology α
⊢ Continuous[g.toTopologicalSpace, g.toTopologicalSpace] Inv.inv

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.GroupTopology

Goal state before:
pp: α : Type u
inst✝ : Group α
f g : GroupTopology α
h : f.toTopologicalSpace = g.toTopologicalSpace
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.GroupTopology

Goal state before:
pp: G : Type w
H : Type x
α : Type u
β : Type v
inst✝ : Group α
_t : TopologicalSpace α := ⊥
⊢ Continuous[instTopologicalSpaceProd, _t] fun p ↦ p.1 * p.2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.GroupTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
t : TopologicalSpace α
inst✝ : Group β
f : α → β
⊢ Continuous[t, (coinduced f).toTopologicalSpace] f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u_1
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : T2Space G
H : Subgroup G
inst✝ : DiscreteTopology ↥H
⊢ IsClosed[inst✝³] ↑H

Tactic:
by

Goal state after: