mathlib-initiative/mathlib-tactics huggingface dataset:
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:
refine
    continuous_induced_rng.2
      (Continuous.prodMk f.continuous
        (MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => ?_)))
  have := NormedRing.inverse_continuousAt (h x).unit
  simp only
  simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢
  exact this.comp (f.continuousAt x)

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:
simpa using

Goal state after:
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
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:
simpa using

Goal state after:
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
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0
c : CU P
hx : x ∈ c.C
⊢ midpoint ℝ (approx n c.left x) (approx n c.right x) = 0

Tactic:
ihn,

Goal state after:
case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0
c : CU P
hx : x ∈ c.C
⊢ midpoint ℝ 0 (approx n c.right x) = 0

case succ.hx
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0
c : CU P
hx : x ∈ c.C
⊢ x ∈ c.left.C
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0
c : CU P
hx : x ∈ c.C
⊢ midpoint ℝ 0 (approx n c.right x) = 0

pp: case succ.hx
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0
c : CU P
hx : x ∈ c.C
⊢ x ∈ c.left.C

Tactic:
ihn,

Goal state after:
case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0
c : CU P
hx : x ∈ c.C
⊢ midpoint ℝ 0 0 = 0

case succ.hx
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0
c : CU P
hx : x ∈ c.C
⊢ x ∈ c.right.C

case succ.hx
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0
c : CU P
hx : x ∈ c.C
⊢ x ∈ c.left.C
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:
simpa using

Goal state after:
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
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1
c : CU P
hx : x ∉ c.U
⊢ midpoint ℝ (approx n c.left x) (approx n c.right x) = 1

Tactic:
ihn,

Goal state after:
case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1
c : CU P
hx : x ∉ c.U
⊢ midpoint ℝ 1 (approx n c.right x) = 1

case succ.hx
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1
c : CU P
hx : x ∉ c.U
⊢ x ∉ c.left.U
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1
c : CU P
hx : x ∉ c.U
⊢ midpoint ℝ 1 (approx n c.right x) = 1

pp: case succ.hx
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1
c : CU P
hx : x ∉ c.U
⊢ x ∉ c.left.U

Tactic:
ihn,

Goal state after:
case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1
c : CU P
hx : x ∉ c.U
⊢ midpoint ℝ 1 1 = 1

case succ.hx
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1
c : CU P
hx : x ∉ c.U
⊢ x ∉ c.right.U

case succ.hx
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1
c : CU P
hx : x ∉ c.U
⊢ x ∉ c.left.U
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
s : Set (WeakDual 𝕜 E)
⊢ Bornology.IsBounded s ↔ IsVonNBounded 𝕜 s

Tactic:
constructor
  · exact fun h => ((NormedSpace.isVonNBounded_iff 𝕜).mpr h).of_topologicalSpace_le
      Dual.dual_norm_topology_le_weak_dual_topology
  · intro h_vN
    have h_ptwise := (withSeminorms 𝕜 E).isVonNBounded_iff_seminorm_bounded.mp h_vN
    obtain ⟨C, hC⟩ := banach_steinhaus (g := fun i : s ↦ WeakDual.toStrongDual i.val) fun x ↦
      let ⟨M, _, hM⟩ := h_ptwise x
      ⟨M, fun i ↦ le_of_lt (hM i.val i.property)⟩
    rw [← isBounded_toWeakDual_preimage_iff_isBounded, isBounded_iff_forall_norm_le]
    exact ⟨C, fun f hf ↦ hC ⟨StrongDual.toWeakDual f, hf⟩⟩

Goal state after:
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SeparableSpace E
inst✝ : ProperSpace 𝕜
s : Set (WeakDual 𝕜 E)
hb : Bornology.IsBounded s
hc : IsClosed s
b_isCompact' : CompactSpace ↑s
b_isMetrizable : MetrizableSpace ↑s
seq_cont_phi : SeqContinuous fun φ ↦ ↑φ
⊢ IsSeqCompact s

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
E : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SeparableSpace E
inst✝ : ProperSpace 𝕜
s : Set (WeakDual 𝕜 E)
hb : Bornology.IsBounded s
hc : IsClosed s
b_isCompact' : CompactSpace ↑s
b_isMetrizable : MetrizableSpace ↑s
seq_cont_phi : SeqContinuous fun φ ↦ ↑φ
⊢ IsSeqCompact s
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: 𝕜✝ : Type u_1
M : Type u_2
E✝ : Type u_3
inst✝¹⁵ : NontriviallyNormedField 𝕜✝
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : TopologicalSpace M
inst✝¹² : Module 𝕜✝ M
inst✝¹¹ : SeminormedAddCommGroup E✝
inst✝¹⁰ : NormedSpace 𝕜✝ E✝
α : Type u_4
𝕜 : Type u_5
E : Type u_6
F : Type u_7
inst✝⁹ : TopologicalSpace α
inst✝⁸ : RCLike 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : TopologicalSpace F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : Module ℝ F
inst✝ : IsScalarTower ℝ 𝕜 F
x : F
⊢ Continuous fun a ↦ re (((↑(toStrongDual ≪≫ₗ extendRCLikeₗ ≪≫ₗ LinearEquiv.restrictScalars ℝ toWeakDual)).toFun a) x)

Tactic:
simpa using

Goal state after:
𝕜✝ : Type u_1
M : Type u_2
E✝ : Type u_3
inst✝¹⁵ : NontriviallyNormedField 𝕜✝
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : TopologicalSpace M
inst✝¹² : Module 𝕜✝ M
inst✝¹¹ : SeminormedAddCommGroup E✝
inst✝¹⁰ : NormedSpace 𝕜✝ E✝
α : Type u_4
𝕜 : Type u_5
E : Type u_6
F : Type u_7
inst✝⁹ : TopologicalSpace α
inst✝⁸ : RCLike 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : TopologicalSpace F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : Module ℝ F
inst✝ : IsScalarTower ℝ 𝕜 F
x : F
⊢ Continuous fun a ↦ a x
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
f : C(X, ℝ≥0)
c : ℝ≥0
hc : 0 < c
x : X
hx : x ∈ {x | c ≤ f x}
⊢ ({ toFun := (⇑f ⊔ ⇑(const X c))⁻¹, continuous_toFun := ⋯ } * f) x = 1 x

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
f : C(X, ℝ≥0)
c : ℝ≥0
hc : 0 < c
x : X
hx : x ∈ {x | c ≤ f x}
⊢ (f x)⁻¹ * f x = 1
Module: Mathlib.Topology.ContinuousMap.ContinuousSqrt

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
x : 𝕜 × 𝕜
hx : 0 ≤ x.2 - x.1
hx' : ↑(re (x.2 - x.1)) = x.2 - x.1
⊢ 0 ≤ re (x.2 - x.1)

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
x : 𝕜 × 𝕜
hx : 0 ≤ x.2 - x.1
hx' : ↑(re (x.2 - x.1)) = x.2 - x.1
⊢ re x.1 ≤ re x.2
Module: Mathlib.Topology.ContinuousMap.ContinuousSqrt

Goal state before:
pp: ⊢ ∀ (x : ℝ≥0 × ℝ≥0), x.1 ≤ x.2 → x.2 = x.1 + (⇑NNReal.sqrt ∘ fun x ↦ x.2 - x.1) x * (⇑NNReal.sqrt ∘ fun x ↦ x.2 - x.1) x

Tactic:
simpa using

Goal state after:
⊢ ∀ (a b : ℝ≥0), a ≤ b → b = a + (b - a)
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
g : C(X, ℝ)
hgs : EqOn (⇑g) 0 s
hgt : EqOn (⇑g) 1 t
hicc : ∀ (x : X), 0 ≤ g x ∧ g x ≤ 1
x : X
⊢ (1 - g) x ∈ Icc 0 1

Tactic:
simpa using

Goal state after:
case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
g : C(X, ℝ)
hgs : EqOn (⇑g) 0 s
hgt : EqOn (⇑g) 1 t
hicc : ∀ (x : X), 0 ≤ g x ∧ g x ≤ 1
x : X
⊢ 0 ≤ g x ∧ g x ≤ 1
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
hx : x ∈ s
⊢ { toFun := fun x ↦ 1 - f x, continuous_toFun := ⋯ } x = 1 x

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
hx : x ∈ s
⊢ f x = 0
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
hx : x ∈ t
⊢ { toFun := fun x ↦ 1 - f x, continuous_toFun := ⋯ } x = 0 x

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
hx : x ∈ t
⊢ 1 = f x
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
⊢ 0 ≤ f x ∧ f x ≤ 1

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
⊢ 0 ≤ f x ∧ f x ≤ 1
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case refine_3.refine_2.inl
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
s₁ s₂ : Set X
g₁ : C(X, ℝ≥0)
hI₁ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ ∈ I
hgt₁ : ∀ x ∈ s₁, 0 < g₁ x
g₂ : C(X, ℝ≥0)
hI₂ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂ ∈ I
hgt₂ : ∀ x ∈ s₂, 0 < g₂ x
x : X
hx : x ∈ s₁
⊢ 0 < (g₁ + g₂) x

Tactic:
simpa using

Goal state after:
case refine_3.refine_2.inl
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
s₁ s₂ : Set X
g₁ : C(X, ℝ≥0)
hI₁ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ ∈ I
hgt₁ : ∀ x ∈ s₁, 0 < g₁ x
g₂ : C(X, ℝ≥0)
hI₂ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂ ∈ I
hgt₂ : ∀ x ∈ s₂, 0 < g₂ x
x : X
hx : x ∈ s₁
⊢ 0 < g₁ x ∨ 0 < g₂ x
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case refine_3.refine_2.inr
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
s₁ s₂ : Set X
g₁ : C(X, ℝ≥0)
hI₁ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ ∈ I
hgt₁ : ∀ x ∈ s₁, 0 < g₁ x
g₂ : C(X, ℝ≥0)
hI₂ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂ ∈ I
hgt₂ : ∀ x ∈ s₂, 0 < g₂ x
x : X
hx : x ∈ s₂
⊢ 0 < (g₁ + g₂) x

Tactic:
simpa using

Goal state after:
case refine_3.refine_2.inr
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
s₁ s₂ : Set X
g₁ : C(X, ℝ≥0)
hI₁ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ ∈ I
hgt₁ : ∀ x ∈ s₁, 0 < g₁ x
g₂ : C(X, ℝ≥0)
hI₂ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂ ∈ I
hgt₂ : ∀ x ∈ s₂, 0 < g₂ x
x : X
hx : x ∈ s₂
⊢ 0 < g₁ x ∨ 0 < g₂ x
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
x : X
hx : x ∈ mᶜ
n : ℕ
⊢ mᶜ ⊆ (U n ∩ interior m)ᶜ

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
x : X
hx : x ∈ mᶜ
n : ℕ
⊢ U n ∩ interior m ⊆ m
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
hI : IsClosed ↑I
hI' : I.IsMaximal
x : X
hx : setOfIdeal I = {x}ᶜ
⊢ idealOfSet 𝕜 {x}ᶜ = I

Tactic:
simpa using

Goal state after:
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
hI : IsClosed ↑I
hI' : I.IsMaximal
x : X
hx : setOfIdeal I = {x}ᶜ
⊢ idealOfSet 𝕜 {x}ᶜ = I
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case h
X : Type u_1
𝕜 : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
inst✝¹ : T2Space X
inst✝ : RCLike 𝕜
x y : X
hxy : x ≠ y
f : C(X, ℝ)
fx : Set.EqOn (⇑f) 0 {x}
fy : Set.EqOn (⇑f) 1 {y}
⊢ ((continuousMapEval X 𝕜) x) ({ toFun := fun x ↦ ↑x, continuous_toFun := ⋯ }.comp f) ≠
    ((continuousMapEval X 𝕜) y) ({ toFun := fun x ↦ ↑x, continuous_toFun := ⋯ }.comp f)

Tactic:
simpa using

Goal state after:
case h
X : Type u_1
𝕜 : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
inst✝¹ : T2Space X
inst✝ : RCLike 𝕜
x y : X
hxy : x ≠ y
f : C(X, ℝ)
fx : Set.EqOn (⇑f) 0 {x}
fy : Set.EqOn (⇑f) 1 {y}
⊢ ¬f x = f y
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case refine_2
X : Type u_1
𝕜 : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
inst✝¹ : T2Space X
inst✝ : RCLike 𝕜
φ : ↑(characterSpace 𝕜 C(X, 𝕜))
x : X
hx : idealOfSet 𝕜 {x}ᶜ = RingHom.ker φ
f : C(X, 𝕜)
⊢ f ∈ RingHom.ker ((continuousMapEval X 𝕜) x) ↔ f ∈ RingHom.ker φ

Tactic:
simpa using

Goal state after:
case refine_2
X : Type u_1
𝕜 : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
inst✝¹ : T2Space X
inst✝ : RCLike 𝕜
φ : ↑(characterSpace 𝕜 C(X, 𝕜))
x : X
hx : idealOfSet 𝕜 {x}ᶜ = RingHom.ker φ
f : C(X, 𝕜)
⊢ f x = 0 ↔ φ f = 0
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
hgmc : EqOn g 0 mᶜ
I : ∀ (n : ℕ) (x : X), u n * (f n) x ≤ u n
S : ∀ (x : X), Summable fun n ↦ u n * (f n) x
n : ℕ
x : X
⊢ ‖u n * (f n) x‖ ≤ u n

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
hgmc : EqOn g 0 mᶜ
I : ∀ (n : ℕ) (x : X), u n * (f n) x ≤ u n
S : ∀ (x : X), Summable fun n ↦ u n * (f n) x
n : ℕ
x : X
⊢ u n * (f n) x ≤ u n
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
hgmc : EqOn g 0 mᶜ
I : ∀ (n : ℕ) (x : X), u n * (f n) x ≤ u n
S : ∀ (x : X), Summable fun n ↦ u n * (f n) x
x : X
hx : x ∈ sᶜ
⊢ ∃ n, x ∉ U n

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
hgmc : EqOn g 0 mᶜ
I : ∀ (n : ℕ) (x : X), u n * (f n) x ≤ u n
S : ∀ (x : X), Summable fun n ↦ u n * (f n) x
x : X
hx : x ∈ sᶜ
⊢ ∃ n, x ∉ U n