mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝⁸ : RingHomIsometric σ₁₂
inst✝⁷ : AddCommGroup E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜₁ E
inst✝ : BaireSpace E
p : Seminorm 𝕜₁ E
hp : LowerSemicontinuous ⇑p
h₁ : ∀ (n : ℕ), IsClosed[inst✝³] (p.closedBall 0 ↑n)
h₂ : ⋃ n, p.closedBall 0 ↑n = univ
n : ℕ
x : E
hxn : {y | p y ≤ ↑n} ∈ map (fun x_1 ↦ x + x_1) (𝓝 0)
hxn' : p x ≤ ↑n
y : E
hy : y ∈ (fun x_1 ↦ x + x_1) ⁻¹' {y | p y ≤ ↑n}
⊢ p y = p (x + y - x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝⁸ : RingHomIsometric σ₁₂
inst✝⁷ : AddCommGroup E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜₁ E
inst✝ : BaireSpace E
⊢ ∀ (p : Seminorm 𝕜₁ E), LowerSemicontinuous ⇑p → Continuous[inst✝³, _] ⇑p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜₁ E
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : UniformSpace E
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup E
inst✝² : IsUniformAddGroup F
inst✝¹ : ContinuousSMul 𝕜₁ E
inst✝ : BarrelledSpace 𝕜₁ E
𝓕 : ι → E →SL[σ₁₂] F
q : SeminormFamily 𝕜₂ F κ
hq : WithSeminorms q
H : ∀ (k : κ) (x : E), BddAbove (range fun i ↦ (q k) ((𝓕 i) x))
k : κ
⊢ BddAbove (range fun i ↦ (q k).comp ↑(𝓕 i))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜₁ E
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : UniformSpace E
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup E
inst✝² : IsUniformAddGroup F
inst✝¹ : ContinuousSMul 𝕜₁ E
inst✝ : BarrelledSpace 𝕜₁ E
𝓕 : ι → E →SL[σ₁₂] F
q : SeminormFamily 𝕜₂ F κ
hq : WithSeminorms q
H : ∀ (k : κ) (x : E), BddAbove (range fun i ↦ (q k) ((𝓕 i) x))
⊢ UniformEquicontinuous (DFunLike.coe ∘ 𝓕)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: ι : Type u_2
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹³ : NontriviallyNormedField 𝕜₁
inst✝¹² : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹¹ : RingHomIsometric σ₁₂
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜₁ E
inst✝⁷ : Module 𝕜₂ F
inst✝⁶ : UniformSpace E
inst✝⁵ : UniformSpace F
inst✝⁴ : IsUniformAddGroup E
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜₁ E
inst✝¹ : BarrelledSpace 𝕜₁ E
𝓕 : ι → E →SL[σ₁₂] F
inst✝ : PolynormableSpace 𝕜₂ F
H : ∀ (x : E), IsVonNBounded 𝕜₂ (range fun i ↦ (𝓕 i) x)
⊢ UniformEquicontinuous (DFunLike.coe ∘ 𝓕)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁷ : NontriviallyNormedField 𝕜₁
inst✝¹⁶ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : Module 𝕜₂ F
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformSpace F
inst✝⁸ : IsUniformAddGroup E
inst✝⁷ : IsUniformAddGroup F
inst✝⁶ : ContinuousSMul 𝕜₁ E
inst✝⁵ : BarrelledSpace 𝕜₁ E
𝓕 : ι → E →SL[σ₁₂] F
inst✝⁴ : PolynormableSpace 𝕜₂ F
inst✝³ : ContinuousSMul 𝕜₂ F
inst✝² : T2Space F
l : Filter α
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
g : α → E →SL[σ₁₂] F
f : E → F
h : Tendsto (fun n x ↦ (g n) x) l (𝓝 f)
⊢ Continuous[inst✝¹⁰.toTopologicalSpace, inst✝⁹.toTopologicalSpace] (linearMapOfTendsto f (fun a ↦ ↑(g a)) h).toFun

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
M : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : C(X, M)ˣ
x : X
⊢ ↑((unitsLift.symm f) x)⁻¹ = ↑f⁻¹ x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
R : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
x : X
this : ContinuousAt Ring.inverse ↑⋯.unit
⊢ ContinuousAt (fun x ↦ ↑((fun x ↦ ⋯.unit) x)⁻¹) x

Tactic:
simp only

Goal state after:
X : Type u_1
R : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
x : X
this : ContinuousAt Ring.inverse ↑⋯.unit
⊢ ContinuousAt (fun x ↦ ↑⋯.unit⁻¹) x
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
R : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
⊢ Continuous fun x ↦ ⋯.unit

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
M : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
⊢ { toFun := fun x ↦ ↑((unitsOfForallIsUnit h) x), continuous_toFun := ⋯ } = f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
R : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
⊢ ↑(unitsLift (unitsOfForallIsUnit h)) = f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
R : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : NormedDivisionRing R
inst✝ : CompleteSpace R
f : C(X, R)
⊢ IsUnit f ↔ ∀ (x : X), f x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
R : Type u_3
𝕜 : Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : NormedField 𝕜
inst✝² : NormedDivisionRing R
inst✝¹ : Algebra 𝕜 R
inst✝ : CompleteSpace R
f : C(X, R)
⊢ spectrum 𝕜 f = ⇑(algebraMap 𝕜 R) ⁻¹' Set.range ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
𝕜 : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : NormedField 𝕜
inst✝ : CompleteSpace 𝕜
f : C(X, 𝕜)
⊢ spectrum 𝕜 f = Set.range ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Basic

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : NontriviallyNormedField K
inst✝¹ : NormedField L
inst✝ : NormedAlgebra K L
F : IntermediateField K L
⊢ ∃ x, 1 < ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Basic

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
⊢ characterSpace 𝕜 A ⊆ ⇑toStrongDual ⁻¹' Metric.closedBall 0 ‖1‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Basic

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
⊢ CompactSpace ↑(characterSpace 𝕜 A)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p₁ p₂ : P
c : 𝕜
⊢ dist p₁ ((homothety p₁ c) p₂) = ‖c‖ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p₁ p₂ : P
c : 𝕜
⊢ dist ((homothety p₁ c) p₂) p₁ = ‖c‖ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p₁ p₂ : P
c₁ c₂ : 𝕜
⊢ dist ((lineMap p₁ p₂) c₁) ((lineMap p₁ p₂) c₂) = dist c₁ c₂ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p₁ p₂ : P
c : 𝕜
⊢ dist ((lineMap p₁ p₂) c) p₁ = ‖c‖ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p₁ p₂ : P
c : 𝕜
⊢ dist ((lineMap p₁ p₂) c) p₂ = ‖1 - c‖ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p₁ p₂ : P
c : 𝕜
⊢ dist ((homothety p₁ c) p₂) p₂ = ‖1 - c‖ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p₁ p₂ : P
c : 𝕜
⊢ dist p₂ ((homothety p₁ c) p₂) = ‖1 - c‖ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist ((lineMap p₁ p₂) ⅟2) p₁ = ‖2‖⁻¹ * dist p₁ p₂

Tactic:
dist_lineMap_left,

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ ‖⅟2‖ * dist p₁ p₂ = ‖2‖⁻¹ * dist p₁ p₂
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist p₁ (midpoint 𝕜 p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist (midpoint 𝕜 p₁ p₂) p₁ = ‖2‖⁻¹ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖⁻¹ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist p₂ (midpoint 𝕜 p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist p₁ (midpoint 𝕜 p₁ p₂) = dist p₂ (midpoint 𝕜 p₁ p₂)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ p₃ p₄ : P
⊢ dist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / ‖2‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p q : P
⊢ dist ((Equiv.pointReflection p) q) p = dist p q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p q : P
⊢ dist ((Equiv.pointReflection p) q) q = ‖2‖ * dist p q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : Q
h : p₁ ≠ p₂
c₁ c₂ : 𝕜
⊢ dist c₁ c₂ ≤ ↑(nndist p₁ p₂)⁻¹ * dist ((lineMap p₁ p₂) c₁) ((lineMap p₁ p₂) c₂)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
inst✝¹ : SeminormedAddCommGroup V
inst✝ : NormedSpace ℝ V
p₁ p₂ p₃ p₄ : V
⊢ dist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
n : ℕ
x : X
hx : x ∈ c.C
⊢ approx n c x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
W : Type u_3
Q : Type u_4
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : PseudoMetricSpace P
inst✝⁵ : NormedAddTorsor V P
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
inst✝¹ : NormedSpace ℝ V
inst✝ : NormedSpace ℝ W
f : P → Q
h : ∀ (x y : P), f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
hfc :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : P := Classical.arbitrary P
⊢ (⇑(AffineEquiv.vaddConst ℝ (f c)).symm ∘ f ∘ ⇑(AffineEquiv.vaddConst ℝ c)) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
n : ℕ
x : X
hx : x ∉ c.U
⊢ approx n c x = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
W : Type u_3
Q : Type u_4
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : PseudoMetricSpace P
inst✝⁵ : NormedAddTorsor V P
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
inst✝¹ : NormedSpace ℝ V
inst✝ : NormedSpace ℝ W
f : P → Q
h : ∀ (x y : P), f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
hfc :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : P := Classical.arbitrary P
x y : V
⊢ (⇑(AffineEquiv.vaddConst ℝ (f c)).symm ∘ f ∘ ⇑(AffineEquiv.vaddConst ℝ c)) (midpoint ℝ x y) =
    midpoint ℝ ((⇑(AffineEquiv.vaddConst ℝ (f c)).symm ∘ f ∘ ⇑(AffineEquiv.vaddConst ℝ c)) x)
      ((⇑(AffineEquiv.vaddConst ℝ (f c)).symm ∘ f ∘ ⇑(AffineEquiv.vaddConst ℝ c)) y)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
n : ℕ
x : X
⊢ 0 ≤ approx n c x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
n : ℕ
x : X
⊢ approx n c x ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
W : Type u_3
Q : Type u_4
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : PseudoMetricSpace P
inst✝⁵ : NormedAddTorsor V P
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
inst✝¹ : NormedSpace ℝ V
inst✝ : NormedSpace ℝ W
f : P → Q
h : ∀ (x y : P), f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
hfc :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : P := Classical.arbitrary P
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ⇑(AddMonoidHom.ofMapMidpoint ℝ ℝ (⇑(AffineEquiv.vaddConst ℝ (f c)).symm ∘ f ∘ ⇑(AffineEquiv.vaddConst ℝ c)) ⋯ ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c₁ c₂ : CU P
h : c₁.U ⊆ c₂.C
n₁ n₂ : ℕ
x : X
⊢ approx n₂ c₂ x ≤ approx n₁ c₁ x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
W : Type u_3
Q : Type u_4
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : PseudoMetricSpace P
inst✝⁵ : NormedAddTorsor V P
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
inst✝¹ : NormedSpace ℝ V
inst✝ : NormedSpace ℝ W
f : P → Q
h : ∀ (x y : P), f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
hfc :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : P := Classical.arbitrary P
p : P
⊢ f p =
    ↑((AddMonoidHom.ofMapMidpoint ℝ ℝ (⇑(AffineEquiv.vaddConst ℝ (f c)).symm ∘ f ∘ ⇑(AffineEquiv.vaddConst ℝ c)) ⋯
                ⋯).toRealLinearMap
            ⋯)
        (p -ᵥ c) +ᵥ
      f c

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a a' b : E
r : k
ha : a ≤ a'
hr : r ≤ 1
⊢ (lineMap a b) r ≤ (lineMap a' b) r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : NormedDivisionRing 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
P : Type u_3
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
k : 𝕜
hk : k ≠ 0
x : E
⊢ (k⁻¹ • fun x ↦ x -ᵥ c) ((fun x ↦ k • x +ᵥ c) x) = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a a' b : E
r : k
ha : a < a'
hr : r < 1
⊢ (lineMap a b) r < (lineMap a' b) r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : NormedDivisionRing 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
P : Type u_3
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
k : 𝕜
hk : k ≠ 0
p : P
⊢ (fun x ↦ k • x +ᵥ c) ((k⁻¹ • fun x ↦ x -ᵥ c) p) = p

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁶ : Ring k
inst✝⁵ : PartialOrder k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b b' : E
r : k
hb : b ≤ b'
hr : 0 ≤ r
⊢ (lineMap a b) r ≤ (lineMap a b') r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : NormedDivisionRing 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
P : Type u_3
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
k : 𝕜
hk : k ≠ 0
x y : E
⊢ edist (k • x +ᵥ c) (k • y +ᵥ c) = ↑‖k‖₊ * edist x y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁶ : Ring k
inst✝⁵ : PartialOrder k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b b' : E
r : k
hb : b < b'
hr : 0 < r
⊢ (lineMap a b) r < (lineMap a b') r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : NormedDivisionRing 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
P : Type u_3
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
k : 𝕜
hk : k ≠ 0
x y : E
h : dist x y ≠ 0
⊢ dist ((smulTorsor c hk) x) ((smulTorsor c hk) y) = ↑‖k‖₊ * dist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : NormedDivisionRing 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
P : Type u_3
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
k : 𝕜
hk : k ≠ 0
⊢ ⇑(smulTorsor c hk) ⁻¹' Metric.ball c ‖k‖ = Metric.ball 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
n : ℕ
x : X
⊢ approx n c x ∈ Icc (approx n c.right x) (approx n c.left x)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a a' b b' : E
r : k
ha : a < a'
hb : b < b'
h₀ : 0 ≤ r
h₁ : r ≤ 1
⊢ (lineMap a b) r < (lineMap a' b') r

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁸ : Ring k
inst✝⁷ : PartialOrder k
inst✝⁶ : IsOrderedRing k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
a b : E
r r' : k
inst✝ : PosSMulReflectLT k E
h : r < r'
⊢ (lineMap a b) r < (lineMap a b) r' ↔ a < b

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁸ : Ring k
inst✝⁷ : PartialOrder k
inst✝⁶ : IsOrderedRing k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
a b : E
r : k
inst✝ : PosSMulReflectLT k E
h : 0 < r
⊢ a < (lineMap a b) r ↔ (lineMap a b) 0 < (lineMap a b) r

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁸ : Ring k
inst✝⁷ : PartialOrder k
inst✝⁶ : IsOrderedRing k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
a b : E
r : k
inst✝ : PosSMulReflectLT k E
h : r < 1
⊢ (lineMap a b) r < b ↔ (lineMap a b) r < (lineMap a b) 1

Tactic:
by

Goal state after: