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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
a : α
⊢ (extensionHom f hf) ↑a = f a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
⊢ IsOpen[_t] ↑v.integer

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁷ : Ring α
inst✝⁶ : UniformSpace α
inst✝⁵ : IsTopologicalRing α
inst✝⁴ : IsUniformAddGroup α
β : Type u
inst✝³ : UniformSpace β
inst✝² : Ring β
inst✝¹ : IsUniformAddGroup β
inst✝ : IsTopologicalRing β
f : α →+* β
hf : Continuous ⇑f
a : α
⊢ (mapRingHom f hf) ↑a = ↑(f a)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
⊢ IsClosed[_t] ↑v.integer

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
⊢ mapRingHom (RingHom.id α) ⋯ = RingHom.id (Completion α)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁷ : Ring α
inst✝⁶ : UniformSpace α
inst✝⁵ : IsTopologicalRing α
inst✝⁴ : IsUniformAddGroup α
β : Type u
inst✝³ : UniformSpace β
inst✝² : Ring β
inst✝¹ : IsUniformAddGroup β
inst✝ : IsTopologicalRing β
f✝ : α →+* β
hf✝ : Continuous ⇑f✝
f : α ≃+* β
hf : Continuous ⇑f
hf' : Continuous ⇑f.symm
⊢ (mapRingHom f.toRingHom hf).comp (mapRingHom f.symm.toRingHom hf') = RingHom.id (Completion β)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁷ : Ring α
inst✝⁶ : UniformSpace α
inst✝⁵ : IsTopologicalRing α
inst✝⁴ : IsUniformAddGroup α
β : Type u
inst✝³ : UniformSpace β
inst✝² : Ring β
inst✝¹ : IsUniformAddGroup β
inst✝ : IsTopologicalRing β
f✝ : α →+* β
hf✝ : Continuous ⇑f✝
f : α ≃+* β
hf : Continuous ⇑f
hf' : Continuous ⇑f.symm
⊢ (mapRingHom f.symm.toRingHom hf').comp (mapRingHom f.toRingHom hf) = RingHom.id (Completion α)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ≠ 0
⊢ (fun x ↦ ↑x⁻¹) = (fun y ↦ ↑y) ∘ fun x ↦ x⁻¹

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ≠ 0
this : (fun x ↦ ↑x⁻¹) = (fun y ↦ ↑y) ∘ fun x ↦ x⁻¹
⊢ 𝓝 0 ⊓ 𝓝 y = ⊥

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝⁶ : Ring A
inst✝⁵ : UniformSpace A
inst✝⁴ : IsUniformAddGroup A
inst✝³ : IsTopologicalRing A
R : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Algebra R A
inst✝ : UniformContinuousConstSMul R A
r : R
⊢ (Completion.map fun x ↦ r • x) = fun x ↦ ↑((algebraMap R A) r) * x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹⁴ : Ring α
inst✝¹³ : UniformSpace α
inst✝¹² : IsTopologicalRing α
inst✝¹¹ : IsUniformAddGroup α
β : Type u
inst✝¹⁰ : UniformSpace β
inst✝⁹ : Ring β
inst✝⁸ : IsUniformAddGroup β
inst✝⁷ : IsTopologicalRing β
f : α →+* β
hf : Continuous ⇑f
A : Type u_2
inst✝⁶ : Ring A
inst✝⁵ : UniformSpace A
inst✝⁴ : IsUniformAddGroup A
inst✝³ : IsTopologicalRing A
R : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Algebra R A
inst✝ : UniformContinuousConstSMul R A
r : R
x : Completion A
a : A
⊢ (coeRingHom.comp (algebraMap R A)) r * ↑a = ↑a * (coeRingHom.comp (algebraMap R A)) r

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹¹ : Ring α
inst✝¹⁰ : UniformSpace α
inst✝⁹ : IsTopologicalRing α
inst✝⁸ : IsUniformAddGroup α
β : Type u
inst✝⁷ : UniformSpace β
inst✝⁶ : Ring β
inst✝⁵ : IsUniformAddGroup β
inst✝⁴ : IsTopologicalRing β
f : α →+* β
hf : Continuous ⇑f
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
inst✝¹ : IsUniformAddGroup R
inst✝ : IsTopologicalRing R
a b : Completion R
⊢ Continuous fun x ↦ x.1 * x.2

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
⊢ ContinuousAt hatInv x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹¹ : Ring α
inst✝¹⁰ : UniformSpace α
inst✝⁹ : IsTopologicalRing α
inst✝⁸ : IsUniformAddGroup α
β : Type u
inst✝⁷ : UniformSpace β
inst✝⁶ : Ring β
inst✝⁵ : IsUniformAddGroup β
inst✝⁴ : IsTopologicalRing β
f : α →+* β
hf : Continuous ⇑f
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
inst✝¹ : IsUniformAddGroup R
inst✝ : IsTopologicalRing R
a b : Completion R
⊢ Continuous fun x ↦ x.2 * x.1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹¹ : Ring α
inst✝¹⁰ : UniformSpace α
inst✝⁹ : IsTopologicalRing α
inst✝⁸ : IsUniformAddGroup α
β : Type u
inst✝⁷ : UniformSpace β
inst✝⁶ : Ring β
inst✝⁵ : IsUniformAddGroup β
inst✝⁴ : IsTopologicalRing β
f : α →+* β
hf : Continuous ⇑f
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
inst✝¹ : IsUniformAddGroup R
inst✝ : IsTopologicalRing R
a✝ b✝ : Completion R
a b : R
⊢ ↑a * ↑b = ↑b * ↑a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹¹ : Ring α
inst✝¹⁰ : UniformSpace α
inst✝⁹ : IsTopologicalRing α
inst✝⁸ : IsUniformAddGroup α
β : Type u
inst✝⁷ : UniformSpace β
inst✝⁶ : Ring β
inst✝⁵ : IsUniformAddGroup β
inst✝⁴ : IsTopologicalRing β
f : α →+* β
hf : Continuous ⇑f
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
inst✝¹ : IsUniformAddGroup R
inst✝ : IsTopologicalRing R
⊢ Algebra R (Completion R)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : IsTopologicalRing α
x y : α
⊢ x - y ∈ closure 0 ↔ x - y ∈ ⊥.closure

Tactic:
by

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

Goal state before:
pp: α✝ : Type u_1
α : Type ?u.4
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : IsTopologicalRing α
x y : α
⊢ (inseparableSetoid α) x y ↔ (Submodule.quotientRel ⊥.closure) x y

Tactic:
by

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

Goal state before:
pp: α✝ : Type u_1
α : Type ?u.4
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : IsTopologicalRing α
⊢ Relator.LiftFun (⇑(inseparableSetoid α)) (⇑(Submodule.quotientRel ⊥.closure)) id id

Tactic:
by

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

Goal state before:
pp: α✝ : Type u_1
α : Type ?u.4
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : IsTopologicalRing α
⊢ Relator.LiftFun (⇑(Submodule.quotientRel ⊥.closure)) (⇑(inseparableSetoid α)) id id

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
⊢ ⋯.extend (⇑f) 1 = 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
h : Continuous (⋯.extend ⇑f)
x y : β
⊢ Continuous fun q ↦ ⋯.extend (⇑f) (q.1 * q.2)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
h : Continuous (⋯.extend ⇑f)
x y : β
⊢ Continuous fun q ↦ ⋯.extend (⇑f) q.1 * ⋯.extend (⇑f) q.2

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
u v : R
⊢ W'.negAddY (u • P) (v • Q) = ((u * v) ^ 2) ^ 3 * W'.negAddY P Q

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : IsTopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
⊢ (↑x)⁻¹ = ↑x⁻¹

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
⊢ ∀ (x y : β), ⋯.extend (⇑f) (x * y) = ⋯.extend (⇑f) x * ⋯.extend (⇑f) y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
x✝¹ x✝ : ValueGroupWithZero R
⊢ ValueGroupWithZero.lift (fun r s ↦ (restrict₀ (ofClass v)) r / (restrict₀ (ofClass v)) ↑s) ⋯ (x✝¹ * x✝) =
    ValueGroupWithZero.lift (fun r s ↦ (restrict₀ (ofClass v)) r / (restrict₀ (ofClass v)) ↑s) ⋯ x✝¹ *
      ValueGroupWithZero.lift (fun r s ↦ (restrict₀ (ofClass v)) r / (restrict₀ (ofClass v)) ↑s) ⋯ x✝

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
⊢ ⋯.extend (⇑f) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
x : R
⊢ (embed v) ((valuation R) x) = (restrict₀ (ofClass v)) x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
γ : ValueGroupWithZero R
⊢ embedding ((embed (valuation R)) γ) = γ

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
f : hat K → hat K := fun x ↦ x * x.hatInv
c : K → hat K := fun x ↦ ↑x
⊢ ContinuousAt f x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
h : Continuous (⋯.extend ⇑f)
x y : β
⊢ Continuous fun q ↦ ⋯.extend (⇑f) (q.1 + q.2)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
h : Continuous (⋯.extend ⇑f)
x y : β
⊢ Continuous fun q ↦ ⋯.extend (⇑f) q.1 + ⋯.extend (⇑f) q.2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
⊢ ∀ (x y : β), ⋯.extend (⇑f) (x + y) = ⋯.extend (⇑f) x + ⋯.extend (⇑f) y

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
f : hat K → hat K := fun x ↦ x * x.hatInv
c : K → hat K := fun x ↦ ↑x
cont : ContinuousAt f x
⊢ x ∈ closure (c '' {0}ᶜ)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
⊢ W'.negAddY P P = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
⊢ StrictMono ⇑(embed v)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
Γ : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝² : v.Compatible
Γ' : Type u_4
inst✝¹ : LinearOrderedCommGroupWithZero Γ'
w : Valuation R Γ'
inst✝ : w.Compatible
x : ValueGroupWithZero R
h : w.IsEquiv v
⊢ h.orderMonoidIso ((embed w) x) = (embed v) x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
x : ValueGroupWithZero R
⊢ (fun x ↦ embedding (⋯.orderMonoidIso x)) ((↑(embed v)).toFun x) = x

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
f : hat K → hat K := fun x ↦ x * x.hatInv
c : K → hat K := fun x ↦ ↑x
cont : ContinuousAt f x
clo : x ∈ closure (c '' {0}ᶜ)
fxclo : f x ∈ closure (f '' c '' {0}ᶜ)
⊢ f '' c '' {0}ᶜ ⊆ {1}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
⊢ Function.Injective fun x ↦ embedding (⋯.orderMonoidIso x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
x : ValueGroupWithZero R
⊢ embedding (⋯.orderMonoidIso ((↑(embed v)).toFun x)) = x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
γ : (ofClass v).ValueGroup₀
x : R
⊢ (orderMonoidIso v) ((valuation R) x) < (orderMonoidIso v) ((orderMonoidIso v).symm γ) ↔ v.restrict x < γ

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
x : hat K
x_ne : x ≠ 0
⊢ x * x.hatInv = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
γ : ValueGroupWithZero R
x : R
⊢ v.restrict x < (orderMonoidIso v) γ ↔ (valuation R) x < γ

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x_ne : x ≠ 0
⊢ x * x⁻¹ = 1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
⊢ 0⁻¹ = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
Γ : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
inst✝³ : LinearOrderedCommGroupWithZero Γ
inst✝² : Nontrivial R
inst✝¹ : NoZeroDivisors R
inst✝ : DecidablePred fun x ↦ x = 0
x : ↥(posSubmonoid R)
⊢ ↑x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁵ : CommSemiring A
inst✝⁴ : Semiring B
inst✝³ : ValuativeRel A
inst✝² : ValuativeRel B
inst✝¹ : Algebra A B
inst✝ : ValuativeExtension A B
a b : A
⊢ (algebraMap A B) a <ᵥ (algebraMap A B) b ↔ a <ᵥ b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁵ : CommSemiring A
inst✝⁴ : Semiring B
inst✝³ : ValuativeRel A
inst✝² : ValuativeRel B
inst✝¹ : Algebra A B
inst✝ : ValuativeExtension A B
x✝ : ↥(posSubmonoid A)
a : A
ha : a ∈ posSubmonoid A
⊢ (algebraMap A B) a ∈ posSubmonoid B

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
x : hat K
x_ne : x ≠ 0
⊢ {0}ᶜ ⊆ {y | y.hatInv = y⁻¹}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁵ : CommSemiring A
inst✝⁴ : Semiring B
inst✝³ : ValuativeRel A
inst✝² : ValuativeRel B
inst✝¹ : Algebra A B
inst✝ : ValuativeExtension A B
⊢ (match 1 with
    | ⟨a, ha⟩ => ⟨(algebraMap A B) a, ⋯⟩) =
    1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
⊢ ∀ ⦃x : hat K⦄, x ≠ 0 → ContinuousAt Inv.inv x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁵ : CommSemiring A
inst✝⁴ : Semiring B
inst✝³ : ValuativeRel A
inst✝² : ValuativeRel B
inst✝¹ : Algebra A B
inst✝ : ValuativeExtension A B
⊢ ∀ (x y : ↥(posSubmonoid A)),
    (match x * y with
      | ⟨a, ha⟩ => ⟨(algebraMap A B) a, ⋯⟩) =
      (match x with
        | ⟨a, ha⟩ => ⟨(algebraMap A B) a, ⋯⟩) *
        match y with
        | ⟨a, ha⟩ => ⟨(algebraMap A B) a, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁷ : CommRing A
inst✝⁶ : Ring B
inst✝⁵ : ValuativeRel A
inst✝⁴ : ValuativeRel B
inst✝³ : Algebra A B
inst✝² : ValuativeExtension A B
Γ : Type u_3
inst✝¹ : LinearOrderedCommMonoidWithZero Γ
w : Valuation B Γ
inst✝ : w.Compatible
⊢ (Valuation.comap (algebraMap A B) w).Compatible

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Ring B
inst✝³ : ValuativeRel A
inst✝² : ValuativeRel B
inst✝¹ : Algebra A B
inst✝ : ValuativeExtension A B
r : A
s : ↥(posSubmonoid A)
⊢ (mapValueGroupWithZero A B) (ValueGroupWithZero.mk r s) =
    ValueGroupWithZero.mk ((algebraMap A B) r) ((mapPosSubmonoid A B) s)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Ring B
inst✝³ : ValuativeRel A
inst✝² : ValuativeRel B
inst✝¹ : Algebra A B
inst✝ : ValuativeExtension A B
a : A
⊢ (mapValueGroupWithZero A B) ((valuation A) a) = (valuation B) ((algebraMap A B) a)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : Ring B
inst✝⁴ : ValuativeRel A
inst✝³ : ValuativeRel B
inst✝² : Algebra A B
inst✝¹ : ValuativeExtension A B
inst✝ : IsRankLeOne B
⊢ IsRankLeOne A

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
inst✝¹ : ValuativeRel R
inst✝ : IsRankLeOne R
⊢ MulArchimedean (ValueGroupWithZero R)

Tactic:
by

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

Goal state before:
pp: K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
F : Filter ↥K
F_cau : Cauchy F
inf_F : 𝓝 0 ⊓ F = ⊥
⊢ Cauchy (map (fun x ↦ x⁻¹) F)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁹ : Field K
inst✝⁸ : UniformSpace K
L✝ : Type u_2
inst✝⁷ : Field L✝
inst✝⁶ : UniformSpace L✝
inst✝⁵ : CompletableTopField L✝
L : Type u_3
inst✝⁴ : Field L
inst✝³ : UniformSpace L
inst✝² : IsTopologicalDivisionRing L
inst✝¹ : T0Space L
inst✝ : CompleteSpace L
F : Filter L
cau_F : Cauchy F
hF : 𝓝 0 ⊓ F = ⊥
this : F.NeBot
x : L
hx : F ≤ 𝓝 x
⊢ x ≠ 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝⁹ : Field K
inst✝⁸ : UniformSpace K
L✝ : Type u_2
inst✝⁷ : Field L✝
inst✝⁶ : UniformSpace L✝
inst✝⁵ : CompletableTopField L✝
L : Type u_3
inst✝⁴ : Field L
inst✝³ : UniformSpace L
inst✝² : IsTopologicalDivisionRing L
inst✝¹ : T0Space L
inst✝ : CompleteSpace L
F : Filter L
cau_F : Cauchy F
hF : 𝓝 0 ⊓ F = ⊥
⊢ Cauchy (map (fun x ↦ x⁻¹) F)

Tactic:
by

Goal state after: