mathlib-initiative/mathlib-tactics huggingface dataset:
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
V : Set G
V_nhds : V ∈ 𝓝 1
V_mul : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U
x y z : G
hz₁ : (x, z) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V
hz₂ : (z, y) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V
⊢ (x, y) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' U

Tactic:
simpa using

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
U : Set G
H : U ∈ 𝓝 1
V : Set G
V_nhds : V ∈ 𝓝 1
V_mul : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U
x y z : G
hz₁ : (x, z) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V
hz₂ : (z, y) ∈ (fun p ↦ p.1⁻¹ * p.2) ⁻¹' V
⊢ x⁻¹ * y ∈ U
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:
simpa using

Goal state after:
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 ∈ L
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:
refine .of_comp isEmbedding_embedProduct ?_
  exact (hf.prodMap (opHomeomorph.isClosedEmbedding.comp
    <| hf.comp opHomeomorph.symm.isClosedEmbedding)).comp isClosedEmbedding_embedProduct

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 : f ×ˢ f ≤ 𝓤 G
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:
uniformity_eq_comap_nhds_one,

Goal state after:
G : Type u_3
inst✝³ : Group G
inst✝² : UniformSpace G
inst✝¹ : IsRightUniformGroup G
inst✝ : WeaklyLocallyCompactSpace G
f : Filter G
hf : f ×ˢ f ≤ comap (fun x ↦ x.2 / x.1) (𝓝 1)
this : f.NeBot
K : Set G
K_compact : IsCompact K
K_mem : K ∈ 𝓝 1
⊢ ∃ x, ∀ᶠ (y : G) in f, y / x ∈ K
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:
simpa using

Goal state after:
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)
⊢ MulOpposite.op x • K ∈ f
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:
simpa using

Goal state after:
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⁻¹) l (𝓝 (x * y⁻¹))
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:
simpa using

Goal state after:
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⁻¹) s
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:
simpa using

Goal state after:
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⁻¹
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:
simpa using

Goal state after:
α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
s : Set α
hs : TotallyBounded s
⊢ TotallyBounded s⁻¹
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:
simpa using

Goal state after:
α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : IsUniformGroup α
s : Set α
hs : TotallyBounded s⁻¹
⊢ TotallyBounded s
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:
simpa using

Goal state after:
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 m.1 x * f' m.1 x, f m.2 x * f' m.2 x) ∈ u
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:
simpa using

Goal state after:
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 m.1 x / f' m.1 x, f m.2 x / f' m.2 x) ∈ u
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:
simpa using

Goal state after:
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 m.1 x)⁻¹, (f m.2 x)⁻¹) ∈ u
Module: Mathlib.Topology.Algebra.GroupWithZero

Goal state before:
pp: case ofNat
G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : TopologicalSpace G₀
inst✝¹ : ContinuousInv₀ G₀
inst✝ : ContinuousMul G₀
x : G₀
m : ℕ
h : x ≠ 0 ∨ 0 ≤ Int.ofNat m
⊢ ContinuousAt (fun x ↦ x ^ Int.ofNat m) x

Tactic:
simpa using

Goal state after:
case ofNat
G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : TopologicalSpace G₀
inst✝¹ : ContinuousInv₀ G₀
inst✝ : ContinuousMul G₀
x : G₀
m : ℕ
h : x ≠ 0 ∨ 0 ≤ Int.ofNat m
⊢ ContinuousAt (fun x ↦ x ^ m) x
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
⊢ UniformSpace.comap (⇑(Equiv.inv G)) (IsTopologicalGroup.leftUniformSpace G) = IsTopologicalGroup.rightUniformSpace G

Tactic:
ext : 1
  change comap (fun (x : G × G) ↦ (Equiv.inv G x.1, Equiv.inv G x.2))
      (comap (fun p : G × G => p.1⁻¹ * p.2) (𝓝 1)) =
    comap (fun p : G × G => p.2 * p.1⁻¹) (𝓝 1)
  have : 𝓝 (1 : G) = comap (Homeomorph.inv G) (𝓝 1) := by rw [Homeomorph.comap_nhds_eq]; simp
  nth_rewrite 1 [this]
  rw [comap_comap, comap_comap]
  simp [Function.comp_def]

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

Goal state before:
pp: G : Type u
inst✝³ : TopologicalSpace G
H : Type v
inst✝² : TopologicalSpace H
inst✝¹ : Mul G
inst✝ : Mul H
e : G ≃* H
he : ∀ (s : Set H), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s
s : Set G
⊢ IsOpen (⇑e.symm ⁻¹' s) ↔ IsOpen s

Tactic:
convert! (he _).symm

Goal state after:
case e'_2
G : Type u
inst✝³ : TopologicalSpace G
H : Type v
inst✝² : TopologicalSpace H
inst✝¹ : Mul G
inst✝ : Mul H
e : G ≃* H
he : ∀ (s : Set H), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s
s : Set G
⊢ s = ⇑e ⁻¹' ⇑e.symm ⁻¹' s
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_4
G : Type u_5
inst✝⁸ : TopologicalSpace α
inst✝⁷ : AddCommGroup α
inst✝⁶ : IsTopologicalAddGroup α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : AddCommGroup β
inst✝³ : TopologicalSpace δ
inst✝² : AddCommGroup δ
inst✝¹ : UniformSpace G
inst✝ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝¹.toTopologicalSpace] fun p ↦ (φ p.1) p.2
W' : Set G
W'_nhds : W' ∈ 𝓝 0
x₀ : α
y₁ : δ
Nx : Filter α := 𝓝 x₀
ee : β × β → α × α := fun u ↦ (e u.1, e u.2)
lim1 : Tendsto (fun a ↦ (a.2 - a.1, y₁)) (comap (⇑e) Nx ×ˢ comap (⇑e) Nx) (𝓝 (0, y₁))
⊢ Tendsto (fun p ↦ (φ p.1) p.2) (𝓝 (0, y₁)) (𝓝 0)

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
δ : Type u_4
G : Type u_5
inst✝⁸ : TopologicalSpace α
inst✝⁷ : AddCommGroup α
inst✝⁶ : IsTopologicalAddGroup α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : AddCommGroup β
inst✝³ : TopologicalSpace δ
inst✝² : AddCommGroup δ
inst✝¹ : UniformSpace G
inst✝ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝¹.toTopologicalSpace] fun p ↦ (φ p.1) p.2
W' : Set G
W'_nhds : W' ∈ 𝓝 0
x₀ : α
y₁ : δ
Nx : Filter α := 𝓝 x₀
ee : β × β → α × α := fun u ↦ (e u.1, e u.2)
lim1 : Tendsto (fun a ↦ (a.2 - a.1, y₁)) (comap (⇑e) Nx ×ˢ comap (⇑e) Nx) (𝓝 (0, y₁))
⊢ Tendsto (fun p ↦ (φ p.1) p.2) (𝓝 (0, y₁)) (𝓝 0)
Module: Mathlib.Topology.Algebra.Ring.Basic

Goal state before:
pp: R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : NonAssocRing R
inst✝ : SeparatelyContinuousMul R
⊢ Continuous[inst✝², inst✝²] fun a ↦ -a

Tactic:
simpa using

Goal state after:
R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : NonAssocRing R
inst✝ : SeparatelyContinuousMul R
⊢ Continuous[inst✝², inst✝²] fun a ↦ -a
Module: Mathlib.Topology.Algebra.IsUniformGroup.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopologicalAddGroup γ
inst✝⁴ : TopologicalSpace δ
inst✝³ : AddCommGroup δ
inst✝² : UniformSpace G
inst✝¹ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝².toTopologicalSpace] fun p ↦ (φ p.1) p.2
W' : Set G
W'_nhds : W' ∈ 𝓝 0
inst✝ : IsUniformAddGroup G
x₀ : α
y₀ : γ
ee : β × β → α × α := fun u ↦ (e u.1, e u.2)
ff : δ × δ → γ × γ := fun u ↦ (f u.1, f u.2)
⊢ Tendsto (fun p ↦ (φ p.1) p.2) (𝓝 (0, 0)) (𝓝 0)

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopologicalAddGroup γ
inst✝⁴ : TopologicalSpace δ
inst✝³ : AddCommGroup δ
inst✝² : UniformSpace G
inst✝¹ : AddCommGroup G
e : β →+ α
de : IsDenseInducing ⇑e
f : δ →+ γ
df : IsDenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous[instTopologicalSpaceProd, inst✝².toTopologicalSpace] fun p ↦ (φ p.1) p.2
W' : Set G
W'_nhds : W' ∈ 𝓝 0
inst✝ : IsUniformAddGroup G
x₀ : α
y₀ : γ
ee : β × β → α × α := fun u ↦ (e u.1, e u.2)
ff : δ × δ → γ × γ := fun u ↦ (f u.1, f u.2)
⊢ Tendsto (fun p ↦ (φ p.1) p.2) (𝓝 (0, 0)) (𝓝 0)
Module: Mathlib.Topology.Algebra.Field

Goal state before:
pp: α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : T1Space 𝕜
inst✝² : Field 𝕜
inst✝¹ : ContinuousInv₀ 𝕜
inst✝ : 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
x : α
hx : x ∈ S
⊢ ((f / g) ^ 2) x = 1 x

Tactic:
simpa using

Goal state after:
α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : T1Space 𝕜
inst✝² : Field 𝕜
inst✝¹ : ContinuousInv₀ 𝕜
inst✝ : 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
x : α
hx : x ∈ S
⊢ f x ^ 2 = g x ^ 2
Module: Mathlib.Topology.Algebra.Field

Goal state before:
pp: α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : T1Space 𝕜
inst✝² : Field 𝕜
inst✝¹ : ContinuousInv₀ 𝕜
inst✝ : 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
hsq : EqOn ((f / g) ^ 2) 1 S
⊢ EqOn f g S ∨ EqOn f (-g) S

Tactic:
simpa using

Goal state after:
α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : T1Space 𝕜
inst✝² : Field 𝕜
inst✝¹ : ContinuousInv₀ 𝕜
inst✝ : 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
hsq : EqOn ((f / g) ^ 2) 1 S
⊢ (∀ ⦃x : α⦄, x ∈ S → f x = g x) ∨ ∀ ⦃x : α⦄, x ∈ S → f x = -g x
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
z : (i : ι) → α i
h₁ : z ∈ univ.pi fun i ↦ Ioc (x i) (update y i₀ m i)
h₂ : z ∈ univ.pi fun i ↦ Ioc (update x i₀ m i) (y i)
⊢ update y i₀ m i₀ < z i₀

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
z : (i : ι) → α i
h₁ : z ∈ univ.pi fun i ↦ Ioc (x i) (update y i₀ m i)
h₂ : z ∈ univ.pi fun i ↦ Ioc (update x i₀ m i) (y i)
⊢ m < z i₀
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' Icc a (f i) = Icc (update f i a) f

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' Icc a (f i) = Icc (update f i a) f
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' Ico a (f i) = Ico (update f i a) f

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' Ico a (f i) = Ico (update f i a) f
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' Ioc a (f i) = Ioc (update f i a) f

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' Ioc a (f i) = Ioc (update f i a) f
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' Ioo a (f i) = Ioo (update f i a) f

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' Ioo a (f i) = Ioo (update f i a) f
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' Icc (f i) b = Icc f (update f i b)

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' Icc (f i) b = Icc f (update f i b)
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' Ico (f i) b = Ico f (update f i b)

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' Ico (f i) b = Ico f (update f i b)
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' Ioc (f i) b = Ioc f (update f i b)

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' Ioc (f i) b = Ioc f (update f i b)
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' Ioo (f i) b = Ioo f (update f i b)

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' Ioo (f i) b = Ioo f (update f i b)
Module: Mathlib.Order.Filter.Interval

Goal state before:
pp: α : Type u_1
l₁ l₁' l₂ l₂' : Filter α
Ixx : α → α → Set α
h : TendstoIxxClass Ixx l₁ l₂
h' : TendstoIxxClass Ixx l₁' l₂'
⊢ Tendsto (fun p ↦ Ixx p.1 p.2) ((l₁ ⊓ l₁') ×ˢ (l₁ ⊓ l₁')) (l₂ ⊓ l₂').smallSets

Tactic:
simpa using

Goal state after:
α : Type u_1
l₁ l₁' l₂ l₂' : Filter α
Ixx : α → α → Set α
h : TendstoIxxClass Ixx l₁ l₂
h' : TendstoIxxClass Ixx l₁' l₂'
⊢ Tendsto (fun p ↦ Ixx p.1 p.2) ((l₁ ⊓ l₁') ×ˢ (l₁ ⊓ l₁')) (l₂.smallSets ⊓ l₂'.smallSets)
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Lattice (α i)
inst✝ : DecidableEq ι
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' uIcc a (f i) = uIcc (update f i a) f

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Lattice (α i)
inst✝ : DecidableEq ι
f : (i : ι) → α i
i : ι
a : α i
⊢ update f i '' uIcc a (f i) = uIcc (update f i a) f
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Lattice (α i)
inst✝ : DecidableEq ι
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' uIcc (f i) b = uIcc f (update f i b)

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Lattice (α i)
inst✝ : DecidableEq ι
f : (i : ι) → α i
i : ι
b : α i
⊢ update f i '' uIcc (f i) b = uIcc f (update f i b)
Module: Mathlib.Order.Interval.Set.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → LinearOrder (α i)
x y z a : (i : ι) → α i
hay : a ∉ univ.pi fun i ↦ Ioc (y i) (z i)
hax : x ≤ a
haz : a ≤ z
⊢ a ∈ ⋃ i, Icc x (update z i (y i))

Tactic:
simpa using

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → LinearOrder (α i)
x y z a : (i : ι) → α i
hay : a ∉ univ.pi fun i ↦ Ioc (y i) (z i)
hax : x ≤ a
haz : a ≤ z
⊢ ∃ i, a i ≤ y i
Module: Mathlib.Topology.Algebra.Order.Group

Goal state before:
pp: case inr.inl
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
this : ∀ {a : G}, Dense (range fun x ↦ a ^ x) → 1 < a → (range fun x ↦ a ^ x) = univ
h : Dense (range fun x ↦ 1 ^ x)
ha₀ : 1 ≤ 1
⊢ (range fun x ↦ 1 ^ x) = univ

Tactic:
simpa using

Goal state after:
case inr.inl
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
this : ∀ {a : G}, Dense (range fun x ↦ a ^ x) → 1 < a → (range fun x ↦ a ^ x) = univ
h : Dense (range fun x ↦ 1 ^ x)
ha₀ : 1 ≤ 1
⊢ {1} = univ
Module: Mathlib.Topology.Algebra.Order.Group

Goal state before:
pp: G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a : G
h : Dense (range fun x ↦ a ^ x)
ha₀ : a ≤ 1
this : ∀ {a : G}, Dense (range fun x ↦ a ^ x) → 1 < a → (range fun x ↦ a ^ x) = univ
hlt : a < 1
⊢ (range fun x ↦ a⁻¹ ^ x) = range fun x ↦ a ^ x

Tactic:
simpa using

Goal state after:
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a : G
h : Dense (range fun x ↦ a ^ x)
ha₀ : a ≤ 1
this : ∀ {a : G}, Dense (range fun x ↦ a ^ x) → 1 < a → (range fun x ↦ a ^ x) = univ
hlt : a < 1
⊢ (range fun x ↦ a⁻¹ ^ x) = range fun x ↦ a ^ x
Module: Mathlib.Topology.Algebra.Order.Group

Goal state before:
pp: G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a : G
h : DenseRange fun x ↦ a ^ x
ha₀ : 1 < a
b : G
hne : (Ioo b (b * a * a)).Nonempty
⊢ ∃ m, a ^ m ∈ Ioo b (b * a * a)

Tactic:
simpa using

Goal state after:
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a : G
h : DenseRange fun x ↦ a ^ x
ha₀ : 1 < a
b : G
hne : (Ioo b (b * a * a)).Nonempty
⊢ ∃ m, b < a ^ m ∧ a ^ m < b * a * a
Module: Mathlib.Topology.Algebra.Order.Group

Goal state before:
pp: G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a : G
h : DenseRange fun x ↦ a ^ x
ha₀ : 1 < a
b : G
m : ℤ
hm : b < a ^ m
hm' : a ^ m < b * a * a
hne : b ≠ a ^ (m - 1)
this : (Ioo (a ^ m) (a ^ (m + 1))).Nonempty
l : ℤ
hl : a ^ l ∈ Ioo (a ^ m) (a ^ (m + 1))
⊢ m < l ∧ l < m + 1

Tactic:
simpa using

Goal state after:
G : Type u_1
inst✝⁴ : TopologicalSpace G
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : OrderTopology G
a : G
h : DenseRange fun x ↦ a ^ x
ha₀ : 1 < a
b : G
m : ℤ
hm : b < a ^ m
hm' : a ^ m < b * a * a
hne : b ≠ a ^ (m - 1)
this : (Ioo (a ^ m) (a ^ (m + 1))).Nonempty
l : ℤ
hl : a ^ l ∈ Ioo (a ^ m) (a ^ (m + 1))
⊢ m < l ∧ l < m + 1