mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case pos.right
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := ⋯
U_nhds_y : ∀ (x y : X), U x y ∈ 𝓝 y
ys : X → Finset X := ⋯
ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := ⋯
ys_nonempty : ∀ (x : X), (ys x).Nonempty
h : X → ↑L := ⋯
lt_h : ∀ (x z : X), f z - ε < ↑(h x) z
h_eq : ∀ (x : X), ↑(h x) x = f x
W : X → Set X := ⋯
W_nhds : ∀ (x : X), W x ∈ 𝓝 x
xs : Finset X := ⋯
xs_w : ⋃ x ∈ xs, W x = ⊤ := ⋯
xs_nonempty : xs.Nonempty
k : ↑L := ⋯
z : X
⊢ f z - ε < ↑k z

Tactic:
·

Goal state after:
case pos.right
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := ⋯
U_nhds_y : ∀ (x y : X), U x y ∈ 𝓝 y
ys : X → Finset X := ⋯
ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := ⋯
ys_nonempty : ∀ (x : X), (ys x).Nonempty
h : X → ↑L := ⋯
lt_h : ∀ (x z : X), f z - ε < ↑(h x) z
h_eq : ∀ (x : X), ↑(h x) x = f x
W : X → Set X := ⋯
W_nhds : ∀ (x : X), W x ∈ 𝓝 x
xs : Finset X := ⋯
xs_w : ⋃ x ∈ xs, W x = ⊤ := ⋯
xs_nonempty : xs.Nonempty
k : ↑L := ⋯
z : X
⊢ f z - ε < ↑k z
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case pos
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
r : ℝ≥0
f : C(X, ℝ)₀
x : X
h : 0 ≤ f x
⊢ ↑((r • f).toNNReal x) = ↑((r • f.toNNReal) x)

pp: case neg
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
r : ℝ≥0
f : C(X, ℝ)₀
x : X
h : f x < 0
⊢ ↑((r • f).toNNReal x) = ↑((r • f.toNNReal) x)

Tactic:
·

Goal state after:
case pos
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
r : ℝ≥0
f : C(X, ℝ)₀
x : X
h : 0 ≤ f x
⊢ ↑((r • f).toNNReal x) = ↑((r • f.toNNReal) x)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case neg
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
r : ℝ≥0
f : C(X, ℝ)₀
x : X
h : f x < 0
⊢ ↑((r • f).toNNReal x) = ↑((r • f.toNNReal) x)

Tactic:
·

Goal state after:
case neg
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
r : ℝ≥0
f : C(X, ℝ)₀
x : X
h : f x < 0
⊢ ↑((r • f).toNNReal x) = ↑((r • f.toNNReal) x)
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : C(X, ℝ)
⊢ f ∈ A.topologicalClosure

Tactic:
rw [subalgebra_topologicalClosure_eq_top_of_separatesPoints A w]
  simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case pos
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 ≤ r
⊢ φ (r • f).toNNReal - φ (-(r • f)).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)

pp: case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : r < 0
⊢ φ (r • f).toNNReal - φ (-(r • f)).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)

Tactic:
·

Goal state after:
case pos
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 ≤ r
⊢ φ (r • f).toNNReal - φ (-(r • f)).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : r < 0
⊢ φ (r • f).toNNReal - φ (-(r • f)).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)

Tactic:
·

Goal state after:
case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : r < 0
⊢ φ (r • f).toNNReal - φ (-(r • f)).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : X → ℝ
c : Continuous[inst✝¹, _] f
ε : ℝ
pos : 0 < ε
g : ↥A
b : ‖↑g - { toFun := f, continuous_toFun := c }‖ < ε
⊢ ∃ g, ∀ (x : X), ‖↑g x - f x‖ < ε

Tactic:
use g

Goal state after:
case h
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : X → ℝ
c : Continuous[inst✝¹, _] f
ε : ℝ
pos : 0 < ε
g : ↥A
b : ‖↑g - { toFun := f, continuous_toFun := c }‖ < ε
⊢ ∀ (x : X), ‖↑g x - f x‖ < ε
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case refine_1
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
hFA : F ∈ A
⊢ { toFun := fun x ↦ ‖F x‖ ^ 2, continuous_toFun := ⋯ } ∈
    ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) (restrictScalars ℝ A.toSubalgebra))

pp: case refine_2
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
hFA : F ∈ A
⊢ (fun f ↦ ⇑f) { toFun := fun x ↦ ‖F x‖ ^ 2, continuous_toFun := ⋯ } x₁ ≠
    (fun f ↦ ⇑f) { toFun := fun x ↦ ‖F x‖ ^ 2, continuous_toFun := ⋯ } x₂

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
hFA : F ∈ A
⊢ { toFun := fun x ↦ ‖F x‖ ^ 2, continuous_toFun := ⋯ } ∈
    ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) (restrictScalars ℝ A.toSubalgebra))
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case refine_2
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
hFA : F ∈ A
⊢ (fun f ↦ ⇑f) { toFun := fun x ↦ ‖F x‖ ^ 2, continuous_toFun := ⋯ } x₁ ≠
    (fun f ↦ ⇑f) { toFun := fun x ↦ ‖F x‖ ^ 2, continuous_toFun := ⋯ } x₂

Tactic:
·

Goal state after:
case refine_2
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
hFA : F ∈ A
⊢ (fun f ↦ ⇑f) { toFun := fun x ↦ ‖F x‖ ^ 2, continuous_toFun := ⋯ } x₁ ≠
    (fun f ↦ ⇑f) { toFun := fun x ↦ ‖F x‖ ^ 2, continuous_toFun := ⋯ } x₂
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case calc_1
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ BddAbove
    ((fun x_1 ↦ ‖((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2) x_1‖) ''
      Metric.closedBall 0 1)

pp: case calc_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ ‖((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2) b‖ =
    ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖

Tactic:
·

Goal state after:
case calc_1
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ BddAbove
    ((fun x_1 ↦ ‖((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2) x_1‖) ''
      Metric.closedBall 0 1)
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case h₁
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
y : E
hy : y ∈ Metric.closedBall 0 1
⊢ ‖↑((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1) y‖ ≤ ‖(x⋆ * x).toProd.1‖

pp: case h₂
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
y : E
hy : y ∈ Metric.closedBall 0 1
⊢ ‖↑((mul 𝕜 E) (x⋆ * x).toProd.2) y‖ ≤ ‖(x⋆ * x).toProd.2‖

Tactic:
·

Goal state after:
case h₁
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
y : E
hy : y ∈ Metric.closedBall 0 1
⊢ ‖↑((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1) y‖ ≤ ‖(x⋆ * x).toProd.1‖
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case h₂
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
y : E
hy : y ∈ Metric.closedBall 0 1
⊢ ‖↑((mul 𝕜 E) (x⋆ * x).toProd.2) y‖ ≤ ‖(x⋆ * x).toProd.2‖

Tactic:
·

Goal state after:
case h₂
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
y : E
hy : y ∈ Metric.closedBall 0 1
⊢ ‖↑((mul 𝕜 E) (x⋆ * x).toProd.2) y‖ ≤ ‖(x⋆ * x).toProd.2‖
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case calc_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ ‖((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2) b‖ =
    ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖

Tactic:
·

Goal state after:
case calc_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ ‖((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2) b‖ =
    ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case refine_1
X : Type u_1
inst✝⁹ : TopologicalSpace X
inst✝⁸ : Zero X
A : Type u_2
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : TopologicalSpace A
inst✝³ : IsSemitopologicalRing A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : T2Space A
s : Set ℝ≥0
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝⁴] ⇑φ
hψ : Continuous[_, inst✝⁴] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
s' : Set ℝ := toReal '' s
e : ↑s ≃ₜ ↑s' := isEmbedding_coe.homeomorphImage s
this : Fact (0 ∈ s')
e0 : e 0 = 0
e0' : e.symm 0 = 0
ξ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hξ : Continuous[_, inst✝⁴] ⇑ξ
ξ' : C(↑s', ℝ)₀ →⋆ₙₐ[ℝ] A :=
  ξ.realContinuousMapZeroOfNNReal.comp (nonUnitalStarAlgHom_precomp ℝ { toContinuousMap := ↑e, map_zero' := e0 })
⊢ Continuous ⇑↑(nonUnitalStarAlgHom_precomp ℝ { toContinuousMap := ↑e, map_zero' := e0 }).toNonUnitalAlgHom

pp: case refine_2
X : Type u_1
inst✝⁹ : TopologicalSpace X
inst✝⁸ : Zero X
A : Type u_2
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : TopologicalSpace A
inst✝³ : IsSemitopologicalRing A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : T2Space A
s : Set ℝ≥0
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝⁴] ⇑φ
hψ : Continuous[_, inst✝⁴] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
s' : Set ℝ := toReal '' s
e : ↑s ≃ₜ ↑s' := isEmbedding_coe.homeomorphImage s
this : Fact (0 ∈ s')
e0 : e 0 = 0
e0' : e.symm 0 = 0
ξ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hξ : Continuous[_, inst✝⁴] ⇑ξ
ξ' : C(↑s', ℝ)₀ →⋆ₙₐ[ℝ] A :=
  ξ.realContinuousMapZeroOfNNReal.comp (nonUnitalStarAlgHom_precomp ℝ { toContinuousMap := ↑e, map_zero' := e0 })
⊢ ξ' (ContinuousMapZero.id s') = ξ (ContinuousMapZero.id s)

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
inst✝⁹ : TopologicalSpace X
inst✝⁸ : Zero X
A : Type u_2
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : TopologicalSpace A
inst✝³ : IsSemitopologicalRing A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : T2Space A
s : Set ℝ≥0
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝⁴] ⇑φ
hψ : Continuous[_, inst✝⁴] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
s' : Set ℝ := toReal '' s
e : ↑s ≃ₜ ↑s' := isEmbedding_coe.homeomorphImage s
this : Fact (0 ∈ s')
e0 : e 0 = 0
e0' : e.symm 0 = 0
ξ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hξ : Continuous[_, inst✝⁴] ⇑ξ
ξ' : C(↑s', ℝ)₀ →⋆ₙₐ[ℝ] A :=
  ξ.realContinuousMapZeroOfNNReal.comp (nonUnitalStarAlgHom_precomp ℝ { toContinuousMap := ↑e, map_zero' := e0 })
⊢ Continuous ⇑↑(nonUnitalStarAlgHom_precomp ℝ { toContinuousMap := ↑e, map_zero' := e0 }).toNonUnitalAlgHom
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x✝ x : Unitization 𝕜 E
h : (algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2 = 0
⊢ ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2‖ ≤
    ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1⋆ + (mul 𝕜 E) x.toProd.2⋆‖

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x✝ x : Unitization 𝕜 E
h : ¬(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2 = 0
⊢ ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2‖ ≤
    ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1⋆ + (mul 𝕜 E) x.toProd.2⋆‖

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x✝ x : Unitization 𝕜 E
h : (algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2 = 0
⊢ ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2‖ ≤
    ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1⋆ + (mul 𝕜 E) x.toProd.2⋆‖
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x✝ x : Unitization 𝕜 E
h : ¬(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2 = 0
⊢ ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2‖ ≤
    ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1⋆ + (mul 𝕜 E) x.toProd.2⋆‖

Tactic:
·

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x✝ x : Unitization 𝕜 E
h : ¬(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2 = 0
⊢ ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2‖ ≤
    ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1⋆ + (mul 𝕜 E) x.toProd.2⋆‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case refine_2
X : Type u_1
inst✝⁹ : TopologicalSpace X
inst✝⁸ : Zero X
A : Type u_2
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : TopologicalSpace A
inst✝³ : IsSemitopologicalRing A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : T2Space A
s : Set ℝ≥0
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝⁴] ⇑φ
hψ : Continuous[_, inst✝⁴] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
s' : Set ℝ := toReal '' s
e : ↑s ≃ₜ ↑s' := isEmbedding_coe.homeomorphImage s
this : Fact (0 ∈ s')
e0 : e 0 = 0
e0' : e.symm 0 = 0
ξ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hξ : Continuous[_, inst✝⁴] ⇑ξ
ξ' : C(↑s', ℝ)₀ →⋆ₙₐ[ℝ] A :=
  ξ.realContinuousMapZeroOfNNReal.comp (nonUnitalStarAlgHom_precomp ℝ { toContinuousMap := ↑e, map_zero' := e0 })
⊢ ξ' (ContinuousMapZero.id s') = ξ (ContinuousMapZero.id s)

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
inst✝⁹ : TopologicalSpace X
inst✝⁸ : Zero X
A : Type u_2
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : TopologicalSpace A
inst✝³ : IsSemitopologicalRing A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : T2Space A
s : Set ℝ≥0
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝⁴] ⇑φ
hψ : Continuous[_, inst✝⁴] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
s' : Set ℝ := toReal '' s
e : ↑s ≃ₜ ↑s' := isEmbedding_coe.homeomorphImage s
this : Fact (0 ∈ s')
e0 : e 0 = 0
e0' : e.symm 0 = 0
ξ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hξ : Continuous[_, inst✝⁴] ⇑ξ
ξ' : C(↑s', ℝ)₀ →⋆ₙₐ[ℝ] A :=
  ξ.realContinuousMapZeroOfNNReal.comp (nonUnitalStarAlgHom_precomp ℝ { toContinuousMap := ↑e, map_zero' := e0 })
⊢ ξ' (ContinuousMapZero.id s') = ξ (ContinuousMapZero.id s)
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: case h₁
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 * y.ofLp.toProd.1‖ ≤ ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.1‖

pp: case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 • y.ofLp.toProd.2 + (y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2)‖ ≤
    ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖ +
      (‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖)

Tactic:
·

Goal state after:
case h₁
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 * y.ofLp.toProd.1‖ ≤ ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.1‖
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 • y.ofLp.toProd.2 + (y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2)‖ ≤
    ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖ +
      (‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖)

Tactic:
·

Goal state after:
case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 • y.ofLp.toProd.2 + (y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2)‖ ≤
    ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖ +
      (‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖)
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: case h₁
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 • y.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖

pp: case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤
    ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖

Tactic:
·

Goal state after:
case h₁
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 • y.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖