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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace M
inst✝⁴ : Mul M
inst✝³ : ContinuousMul M
inst✝² : TopologicalSpace N
inst✝¹ : Mul N
inst✝ : ContinuousMul N
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun p ↦ p.1 * p.2

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M✝ : Type u_3
N✝ : Type u_4
X : Type u_5
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace M✝
inst✝⁷ : Mul M✝
inst✝⁶ : ContinuousMul M✝
M : Type u_6
N : Type u_7
inst✝⁵ : TopologicalSpace M
inst✝⁴ : Mul M
inst✝³ : SeparatelyContinuousMul M
inst✝² : TopologicalSpace N
inst✝¹ : Mul N
inst✝ : SeparatelyContinuousMul N
x✝ : M × N
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun x ↦ x✝ * x

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M✝ : Type u_3
N✝ : Type u_4
X : Type u_5
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace M✝
inst✝⁷ : Mul M✝
inst✝⁶ : ContinuousMul M✝
M : Type u_6
N : Type u_7
inst✝⁵ : TopologicalSpace M
inst✝⁴ : Mul M
inst✝³ : SeparatelyContinuousMul M
inst✝² : TopologicalSpace N
inst✝¹ : Mul N
inst✝ : SeparatelyContinuousMul N
x✝ : M × N
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun x ↦ x * x✝

Tactic:
by

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

Goal state before:
pp: M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x * x₀) (𝓝 1)
x₀ y₀ : M
⊢ (fun p ↦ x₀ * p.1 * (p.2 * y₀)) = ((fun x ↦ x₀ * x) ∘ fun x ↦ x * y₀) ∘ uncurry fun x1 x2 ↦ x1 * x2

Tactic:
by

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

Goal state before:
pp: M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p ↦ x₀ * p.1 * (p.2 * y₀)) = ((fun x ↦ x₀ * x) ∘ fun x ↦ x * y₀) ∘ uncurry fun x1 x2 ↦ x1 * x2
⊢ ((fun x ↦ x₀ * x) ∘ fun x ↦ y₀ * x) = fun x ↦ x₀ * y₀ * x

Tactic:
by

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

Goal state before:
pp: M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p ↦ x₀ * p.1 * (p.2 * y₀)) = ((fun x ↦ x₀ * x) ∘ fun x ↦ x * y₀) ∘ uncurry fun x1 x2 ↦ x1 * x2
key₂ : ((fun x ↦ x₀ * x) ∘ fun x ↦ y₀ * x) = fun x ↦ x₀ * y₀ * x
⊢ map (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 (x₀, y₀)) = map (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 x₀ ×ˢ 𝓝 y₀)

Tactic:
by

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

Goal state before:
pp: M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p ↦ x₀ * p.1 * (p.2 * y₀)) = ((fun x ↦ x₀ * x) ∘ fun x ↦ x * y₀) ∘ uncurry fun x1 x2 ↦ x1 * x2
key₂ : ((fun x ↦ x₀ * x) ∘ fun x ↦ y₀ * x) = fun x ↦ x₀ * y₀ * x
⊢ map (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 x₀ ×ˢ 𝓝 y₀) = map (fun p ↦ x₀ * p.1 * (p.2 * y₀)) (𝓝 1 ×ˢ 𝓝 1)

Tactic:
by

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

Goal state before:
pp: M : Type u_1
α : Type u_2
β : Type u_3
Γ : Type u_4
inst✝⁶ : Group Γ
T : Type u_5
inst✝⁵ : TopologicalSpace T
inst✝⁴ : MulAction Γ T
inst✝³ : T2Space T
inst✝² : LocallyCompactSpace T
inst✝¹ : ContinuousConstSMul Γ T
inst✝ : ProperlyDiscontinuousSMul Γ T
⊢ T2Space (Quotient (MulAction.orbitRel Γ T))

Tactic:
by

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

Goal state before:
pp: α : Type u_2
G₀ : Type u_6
inst✝⁴ : GroupWithZero G₀
inst✝³ : AddMonoid α
inst✝² : DistribMulAction G₀ α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousConstSMul G₀ α
s : Set α
c : G₀
hc : c ≠ 0
⊢ c • s ∈ 𝓝 0 ↔ s ∈ 𝓝 0

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
ι : Sort u_4
us' : ι → UniformSpace G
h' : ∀ (i : ι), IsUniformGroup G
⊢ IsUniformGroup G

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
u₁ u₂ : UniformSpace G
h₁ : IsUniformGroup G
h₂ : IsUniformGroup G
⊢ IsUniformGroup G

Tactic:
by

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

Goal state before:
pp: G : Type u_1
H : Type u_2
hom : Type u_3
inst✝⁶ : Group G
inst✝⁵ : Group H
inst✝⁴ : UniformSpace G
inst✝³ : UniformSpace H
inst✝² : IsUniformGroup H
inst✝¹ : FunLike hom G H
inst✝ : MonoidHomClass hom G H
f : hom
hf : IsUniformInducing ⇑f
⊢ UniformContinuous fun p ↦ p.1 / p.2

Tactic:
by

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

Goal state before:
pp: G : Type u_1
H : Type u_2
hom : Type u_3
inst✝³ : Group G
inst✝² : Group H
inst✝¹ : UniformSpace G
hG : IsUniformGroup G
inst✝ : UniformSpace H
hH : IsUniformGroup H
⊢ IsUniformGroup (G × H)

Tactic:
by

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

Goal state before:
pp: G✝ : Type u_1
H : Type u_2
hom : Type u_3
inst✝⁴ : Group G✝
inst✝³ : Group H
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → UniformSpace (G i)
inst✝¹ : (i : ι) → Group (G i)
inst✝ : ∀ (i : ι), IsUniformGroup (G i)
⊢ IsUniformGroup ((i : ι) → G i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
h : IsProperMap f
⊢ IsClosedMap f

Tactic:
by

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

Goal state before:
pp: M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p ↦ x₀ * p.1 * (p.2 * y₀)) = ((fun x ↦ x₀ * x) ∘ fun x ↦ x * y₀) ∘ uncurry fun x1 x2 ↦ x1 * x2
key₂ : ((fun x ↦ x₀ * x) ∘ fun x ↦ y₀ * x) = fun x ↦ x₀ * y₀ * x
⊢ map (fun p ↦ x₀ * p.1 * (p.2 * y₀)) (𝓝 1 ×ˢ 𝓝 1) =
    map ((fun x ↦ x₀ * x) ∘ fun x ↦ x * y₀) (map (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
⊢ IsProperMap f ↔
    Continuous[inst✝¹, inst✝] f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x

Tactic:
by

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

Goal state before:
pp: M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p ↦ x₀ * p.1 * (p.2 * y₀)) = ((fun x ↦ x₀ * x) ∘ fun x ↦ x * y₀) ∘ uncurry fun x1 x2 ↦ x1 * x2
key₂ : ((fun x ↦ x₀ * x) ∘ fun x ↦ y₀ * x) = fun x ↦ x₀ * y₀ * x
⊢ map ((fun x ↦ x₀ * x) ∘ fun x ↦ x * y₀) (𝓝 1) = 𝓝 (x₀ * y₀)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
hg : IsProperMap g
hf : IsProperMap f
⊢ Continuous[inst✝², inst✝] (g ∘ f)

Tactic:
by

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

Goal state before:
pp: M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x * x₀) (𝓝 1)
⊢ Continuous[instTopologicalSpaceProd, inst✝] fun p ↦ p.1 * p.2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
hg : IsProperMap g
hf : IsProperMap f
⊢ IsProperMap (g ∘ f)

Tactic:
by

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

Goal state before:
pp: M : Type u
inst✝¹ : CommMonoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x1 x2 ↦ x1 * x2) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x ↦ x₀ * x) (𝓝 1)
⊢ ContinuousMul M

Tactic:
by

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

Goal state before:
pp: M₁ : Type u_6
M₂ : Type u_7
inst✝⁴ : TopologicalSpace M₂
inst✝³ : T2Space M₂
inst✝² : Mul M₁
inst✝¹ : Mul M₂
inst✝ : ContinuousMul M₂
x y : M₁
⊢ Continuous[Pi.topologicalSpace, inst✝⁴] fun x_1 ↦ x_1 x * x_1 y

Tactic:
by

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

Goal state before:
pp: M₁ : Type u_6
M₂ : Type u_7
inst✝⁴ : TopologicalSpace M₂
inst✝³ : T2Space M₂
inst✝² : Mul M₁
inst✝¹ : Mul M₂
inst✝ : ContinuousMul M₂
⊢ IsClosed[Pi.topologicalSpace] {f | ∀ (x y : M₁), f (x * y) = f x * f y}

Tactic:
by

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

Goal state before:
pp: M : Type u_6
N : Type u_7
F : Type u_8
inst✝⁶ : Mul M
inst✝⁵ : Mul N
inst✝⁴ : FunLike F M N
inst✝³ : MulHomClass F M N
inst✝² : TopologicalSpace M
inst✝¹ : TopologicalSpace N
inst✝ : SeparatelyContinuousMul N
f : F
hf : IsInducing ⇑f
m : M
⊢ ∀ {c x : Mᵐᵒᵖ},
    ((⇑opHomeomorph ∘ ⇑f) ∘ ⇑opHomeomorph.symm) (c • x) =
      op (f (unop c)) • ((⇑opHomeomorph ∘ ⇑f) ∘ ⇑opHomeomorph.symm) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
hf : Continuous[inst✝², inst✝¹] f
hg : Continuous[inst✝¹, inst✝] g
hgf : IsProperMap (g ∘ f)
f_surj : Surjective f
⊢ IsProperMap g

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
K U : Set M
hK : IsCompact K
hU : U ∈ 𝓝 0
⊢ univ ∈ 𝓝 0 ∧ ∅ * univ ⊆ U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
hf : Continuous[inst✝², inst✝¹] f
hg : Continuous[inst✝¹, inst✝] g
hgf : IsProperMap (g ∘ f)
g_inj : Injective g
⊢ IsProperMap f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
f : X → Y
g : Y → Z
inst✝ : T2Space Y
hf : Continuous[inst✝³, inst✝²] f
hg : Continuous[inst✝², inst✝¹] g
hgf : IsProperMap (g ∘ f)
⊢ IsProperMap f

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
K U : Set M
hK : IsCompact K
hU : U ∈ 𝓝 0
⊢ ∃ V ∈ 𝓝 0, K * V ⊆ U

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
l : Filter (M × M)
hl : Disjoint l (cocompact (M × M))
⊢ Tendsto (fun x ↦ x.1 * x.2) ((𝓝 0).coprod (𝓝 0) ⊓ l) (𝓝 0)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
l : Filter (M × M)
hl : Disjoint l (cocompact (M × M))
h'l : l ≤ (𝓝 0).coprod (𝓝 0)
⊢ Tendsto (fun x ↦ x.1 * x.2) l (𝓝 0)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
α : Type u_6
β : Type u_7
inst✝⁴ : TopologicalSpace M
inst✝³ : MulZeroClass M
inst✝² : ContinuousMul M
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → M
g : β → M
f_cont : Continuous[inst✝¹, inst✝⁴] f
g_cont : Continuous[inst✝, inst✝⁴] g
hf : Tendsto f (cocompact α) (𝓝 0)
hg : Tendsto g (cocompact β) (𝓝 0)
l : Filter (M × M) := map (Prod.map f g) (cocompact (α × β))
l_def : l = map (Prod.map f g) (cocompact (α × β))
K : Set (M × M) := insert 0 (range f) ×ˢ insert 0 (range g)
K_compact : IsCompact K
K_mem_l : K ∈ l
⊢ Disjoint l (cocompact (M × M))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Z → W
hf : IsProperMap f
hg : IsProperMap g
⊢ IsProperMap (Prod.map f g)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
α : Type u_6
β : Type u_7
inst✝⁴ : TopologicalSpace M
inst✝³ : MulZeroClass M
inst✝² : ContinuousMul M
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → M
g : β → M
f_cont : Continuous[inst✝¹, inst✝⁴] f
g_cont : Continuous[inst✝, inst✝⁴] g
hf : Tendsto f (cocompact α) (𝓝 0)
hg : Tendsto g (cocompact β) (𝓝 0)
l : Filter (M × M) := map (Prod.map f g) (cocompact (α × β))
l_def : l = map (Prod.map f g) (cocompact (α × β))
K : Set (M × M) := insert 0 (range f) ×ˢ insert 0 (range g)
K_compact : IsCompact K
K_mem_l : K ∈ l
l_compact : Disjoint l (cocompact (M × M))
⊢ l ≤ (𝓝 0).coprod (𝓝 0)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
α : Type u_6
β : Type u_7
inst✝⁴ : TopologicalSpace M
inst✝³ : MulZeroClass M
inst✝² : ContinuousMul M
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → M
g : β → M
f_cont : Continuous[inst✝¹, inst✝⁴] f
g_cont : Continuous[inst✝, inst✝⁴] g
hf : Tendsto f (cocompact α) (𝓝 0)
hg : Tendsto g (cocompact β) (𝓝 0)
⊢ Tendsto (fun i ↦ f i.1 * g i.2) (cocompact (α × β)) (𝓝 0)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
α : Type u_6
β : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
f : α → M
g : β → M
hf : Tendsto f cofinite (𝓝 0)
hg : Tendsto g cofinite (𝓝 0)
⊢ Tendsto (fun i ↦ f i.1 * g i.2) cofinite (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: ι : Type u_5
X : ι → Type u_6
Y : ι → Type u_7
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : (i : ι) → TopologicalSpace (Y i)
f : (i : ι) → X i → Y i
h :
  ∀ (i : ι),
    Continuous[inst✝¹ i, inst✝ i] (f i) ∧
      ∀ ⦃𝒰 : Ultrafilter (X i)⦄ ⦃y : Y i⦄, Tendsto (f i) (↑𝒰) (𝓝 y) → ∃ x, f i x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter ((i : ι) → X i)
y : (i : ι) → Y i
hy : Tendsto (fun x i ↦ f i (x i)) (↑𝒰) (𝓝 y)
⊢ ∀ (i : ι), Tendsto (f i) (↑(Ultrafilter.map (eval i) 𝒰)) (𝓝 (y i))

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝⁴ : GroupWithZero M
inst✝³ : TopologicalSpace M
inst✝² : ContinuousMul M
inst✝¹ : CompactSpace M
inst✝ : T1Space M
U : Set M
hU : IsOpen[inst✝³] U
h0U : 0 ∈ U
h1U : 1 ∉ U
W : Set M
hW : W ∈ 𝓝 0
hW' : univ * W ⊆ U
x : M
hx : x ≠ 0
hxW : x ∈ W
⊢ 1 ∈ univ * W

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝⁴ : GroupWithZero M
inst✝³ : TopologicalSpace M
inst✝² : ContinuousMul M
inst✝¹ : CompactSpace M
inst✝ : T1Space M
U : Set M
hU : IsOpen[inst✝³] U
h0U : 0 ∈ U
h1U : 1 ∉ U
W : Set M
hW : W ∈ 𝓝 0
hW' : univ * W ⊆ U
H : ¬∃ x, x ≠ 0 ∧ x ∈ W
⊢ W ⊆ {0}

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝⁴ : GroupWithZero M
inst✝³ : TopologicalSpace M
inst✝² : ContinuousMul M
inst✝¹ : CompactSpace M
inst✝ : T1Space M
U : Set M
hU : IsOpen[inst✝³] U
h0U : 0 ∈ U
h1U : 1 ∉ U
W : Set M
hW : W ∈ 𝓝 0
hW' : univ * W ⊆ U
H : ¬∃ x, x ≠ 0 ∧ x ∈ W
⊢ {0} ⊆ W

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝⁴ : GroupWithZero M
inst✝³ : TopologicalSpace M
inst✝² : ContinuousMul M
inst✝¹ : CompactSpace M
inst✝ : T1Space M
⊢ IsOpen[inst✝³] {0}

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulOneClass M
inst✝ : ContinuousMul M
s : Set M
hs : s ∈ 𝓝 1
⊢ s ∈ 𝓝 (1 * 1)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulOneClass M
inst✝ : ContinuousMul M
s : Set M
hs : s ∈ 𝓝 1
⊢ ∃ V, IsOpen[inst✝²] V ∧ 1 ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulOneClass M
inst✝ : ContinuousMul M
U : Set M
hU : U ∈ 𝓝 1
⊢ ∃ V, IsOpen[inst✝²] V ∧ 1 ∈ V ∧ V * V ⊆ U

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulOneClass M
inst✝ : ContinuousMul M
p : ι → Prop
s : ι → Set M
h : (𝓝 1).HasBasis p s
⊢ (𝓝 1).HasBasis p fun i ↦ s i * s i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: ι : Type u_5
X : ι → Type u_6
Y : ι → Type u_7
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : (i : ι) → TopologicalSpace (Y i)
f : (i : ι) → X i → Y i
h : ∀ (i : ι), IsProperMap (f i)
⊢ IsProperMap fun x i ↦ f i (x i)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M
inst✝² : Semigroup M
inst✝¹ : SeparatelyContinuousMul M
inst✝ : T2Space M
s : Subsemigroup M
hs : ∀ (x y : ↥s), x * y = y * x
this : ∀ x ∈ s, ∀ y ∈ s, x * y = y * x
x✝¹ x✝ : ↥s.topologicalClosure
x : M
hx : x ∈ s.topologicalClosure
y : M
hy : y ∈ s.topologicalClosure
⊢ ↑(⟨x, hx⟩ * ⟨y, hy⟩) = ↑(⟨y, hy⟩ * ⟨x, hx⟩)

Tactic:
by

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

Goal state before:
pp: N : Type u_4
inst✝² : TopologicalSpace N
inst✝¹ : Mul N
inst✝ : ContinuousMul N
s t : Set N
hs : IsCompact s
ht : IsCompact t
⊢ IsCompact (s * t)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : SeparatelyContinuousMul M
a b : M
ha : b * a = 1
⊢ Tendsto (fun x ↦ a * x) (cocompact M) (cocompact M)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : SeparatelyContinuousMul M
a b : M
ha : a * b = 1
⊢ Tendsto (fun x ↦ x) (cocompact M) (cocompact M)

Tactic:
exact Filter.tendsto_id

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : SeparatelyContinuousMul M
a b : M
ha : a * b = 1
⊢ Tendsto (fun x ↦ x * a) (cocompact M) (cocompact M)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
u : Set M
hu : u ∈ 𝓝 1
⊢ ∃ V ∈ 𝓝 1, ∀ {v w s t : M}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → α → M
x : Filter α
a : ι → M
x✝ : ∀ i ∈ [], Tendsto (f i) x (𝓝 (a i))
⊢ Tendsto (fun b ↦ (List.map (fun c ↦ f c b) []).prod) x (𝓝 (List.map a []).prod)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f✝ : ι → α → M
x : Filter α
a : ι → M
f : ι
l : List ι
h : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))
⊢ Tendsto (fun b ↦ (List.map (fun c ↦ f✝ c b) (f :: l)).prod) x (𝓝 (List.map a (f :: l)).prod)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Maps.Proper.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
h : IsProperMap f
K : Set Y
hK : IsCompact K
⊢ IsCompact (f ⁻¹' K)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
M : Type u_3
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → X → M
l : List ι
t : Set X
h : ∀ i ∈ l, ContinuousOn (f i) t
⊢ ContinuousOn (fun a ↦ (List.map (fun i ↦ f i a) l).prod) t

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
⊢ Continuous[inst✝², inst✝²] fun a ↦ a ^ 0

Tactic:
by

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

Goal state before:
pp: M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
k : ℕ
⊢ Continuous[inst✝², inst✝²] fun a ↦ a ^ (k + 1)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Monoid M
inst✝⁵ : ContinuousMul M
R : Type u_6
A : Type u_7
inst✝⁴ : Monoid A
inst✝³ : SMul R A
inst✝² : IsScalarTower R A A
inst✝¹ : TopologicalSpace A
inst✝ : SeparatelyContinuousMul A
q : R
⊢ Continuous[inst✝¹, inst✝¹] fun x ↦ q • x

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Monoid M
inst✝⁵ : ContinuousMul M
R : Type u_6
A : Type u_7
inst✝⁴ : Monoid A
inst✝³ : SMul R A
inst✝² : SMulCommClass R A A
inst✝¹ : TopologicalSpace A
inst✝ : SeparatelyContinuousMul A
q : R
⊢ Continuous[inst✝¹, inst✝¹] fun x ↦ q • x

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → α → M
x : Filter α
a : ι → M
s : Multiset ι
⊢ (∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
    Tendsto (fun b ↦ (Multiset.map (fun c ↦ f c b) s).prod) x (𝓝 (Multiset.map a s).prod)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
M : Type u_3
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
s : Multiset ι
⊢ (∀ i ∈ s, Continuous[inst✝³, inst✝²] (f i)) → Continuous[inst✝³, inst✝²] fun a ↦ (Multiset.map (fun i ↦ f i a) s).prod

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
M : Type u_3
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
s : Multiset ι
t : Set X
⊢ (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a ↦ (Multiset.map (fun i ↦ f i a) s).prod) t

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
X : Type u_6
M : Type u_7
inst✝ : CommMonoid M
s : Finset ι
l : Filter X
f g : ι → X → M
hs : ∀ i ∈ s, f i =ᶠ[l] g i
⊢ ∀ᶠ (x : X) in l, ∀ i ∈ s, f i x = g i x

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
X : Type u_6
M : Type u_7
inst✝ : CommMonoid M
s : Finset ι
l : Filter X
f g : ι → X → M
hs : ∀ i ∈ s, f i =ᶠ[l] g i
⊢ ∏ i ∈ s, f i =ᶠ[l] ∏ i ∈ s, g i

Tactic:
by

Goal state after: