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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
E₁ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (E₁ i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝ : (i : ι) → Module 𝕜 (E₁ i)
f : (i : ι) → E i →L[𝕜] E₁ i
aux : ContinuousMultilinearMap 𝕜 E₁ F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E F :=
  { toFun := fun g ↦ g.compContinuousLinearMap f, map_add' := ⋯, map_smul' := ⋯ }
⊢ ∀ (ib : Set ((i : ι) → E i) × Set F),
    IsVonNBounded 𝕜 ib.1 ∧ ib.2 ∈ 𝓝 0 →
      ∃ ia, (IsVonNBounded 𝕜 ia.1 ∧ ia.2 ∈ 𝓝 0) ∧ ∀ x ∈ {f | MapsTo (⇑f) ia.1 ia.2}, aux x ∈ {f | MapsTo (⇑f) ib.1 ib.2}

Tactic:
]

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
E₁ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (E₁ i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝ : (i : ι) → Module 𝕜 (E₁ i)
f : (i : ι) → E i →L[𝕜] E₁ i
aux : ContinuousMultilinearMap 𝕜 E₁ F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E F :=
  { toFun := fun g ↦ g.compContinuousLinearMap f, map_add' := ⋯, map_smul' := ⋯ }
⊢ ∀ (ib : Set ((i : ι) → E i) × Set F),
    IsVonNBounded 𝕜 ib.1 ∧ ib.2 ∈ 𝓝 0 →
      ∃ ia, (IsVonNBounded 𝕜 ia.1 ∧ ia.2 ∈ 𝓝 0) ∧ ∀ x ∈ {f | MapsTo (⇑f) ia.1 ia.2}, aux x ∈ {f | MapsTo (⇑f) ib.1 ib.2}
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
g : F →L[𝕜] G
aux₁ : ContinuousMultilinearMap 𝕜 E F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E G :=
  { toFun := g.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯ }
⊢ ∀ (ib : Set ((i : ι) → E i) × Set G),
    IsVonNBounded 𝕜 ib.1 ∧ ib.2 ∈ 𝓝 0 →
      ∃ ia,
        (IsVonNBounded 𝕜 ia.1 ∧ ia.2 ∈ 𝓝 0) ∧ ∀ x ∈ {f | MapsTo (⇑f) ia.1 ia.2}, aux₁ x ∈ {f | MapsTo (⇑f) ib.1 ib.2}

Tactic:
]

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
g : F →L[𝕜] G
aux₁ : ContinuousMultilinearMap 𝕜 E F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E G :=
  { toFun := g.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯ }
⊢ ∀ (ib : Set ((i : ι) → E i) × Set G),
    IsVonNBounded 𝕜 ib.1 ∧ ib.2 ∈ 𝓝 0 →
      ∃ ia,
        (IsVonNBounded 𝕜 ia.1 ∧ ia.2 ∈ 𝓝 0) ∧ ∀ x ∈ {f | MapsTo (⇑f) ia.1 ia.2}, aux₁ x ∈ {f | MapsTo (⇑f) ib.1 ib.2}
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsQuotientMap (rangeFactorization f)
⊢ Continuous[inst✝¹, inst✝] f

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsQuotientMap (rangeFactorization f)
⊢ Continuous[inst✝¹, inst✝] f
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
ho : IsOpenMap f
h_cont : Continuous[inst✝¹, inst✝] f
⊢ IsQuotientMap (rangeFactorization f)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
ho : IsOpenMap f
h_cont : Continuous[inst✝¹, inst✝] f
⊢ IsQuotientMap (rangeFactorization f)
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hc : IsClosedMap f
h_cont : Continuous[inst✝¹, inst✝] f
⊢ IsQuotientMap (rangeFactorization f)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hc : IsClosedMap f
h_cont : Continuous[inst✝¹, inst✝] f
⊢ IsQuotientMap (rangeFactorization f)
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : IsStrictMap f ∧ Surjective f
f_strict : IsQuotientMap (rangeFactorization f)
f_surj : Surjective f
⊢ IsQuotientMap f

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : IsStrictMap f ∧ Surjective f
f_strict : IsQuotientMap (rangeFactorization f)
f_surj : Surjective f
⊢ IsQuotientMap f
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : IsStrictMap f ∧ Injective f
f_strict : IsEmbedding (kerLift f)
f_inj : Injective f
⊢ IsEmbedding f

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
x✝ : IsStrictMap f ∧ Injective f
f_strict : IsEmbedding (kerLift f)
f_inj : Injective f
⊢ IsEmbedding f
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
aux : (F →L[𝕜] G) →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E F →L[𝕜] ContinuousMultilinearMap 𝕜 E G :=
  { toFun := fun g ↦ { toFun := g.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ },
    map_add' := ⋯, map_smul' := ⋯ }
⊢ ∀ (ib : Set (ContinuousMultilinearMap 𝕜 E F) × Set ((i : ι) → E i) × Set G),
    IsVonNBounded 𝕜 ib.1 ∧ IsVonNBounded 𝕜 ib.2.1 ∧ ib.2.2 ∈ 𝓝 0 →
      ∃ ia,
        (IsVonNBounded 𝕜 ia.1 ∧ ia.2 ∈ 𝓝 0) ∧
          ∀ x ∈ {f | ∀ x ∈ ia.1, f x ∈ ia.2}, aux x ∈ {f | ∀ x ∈ ib.1, f x ∈ {f | MapsTo (⇑f) ib.2.1 ib.2.2}}

Tactic:
]

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
aux : (F →L[𝕜] G) →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E F →L[𝕜] ContinuousMultilinearMap 𝕜 E G :=
  { toFun := fun g ↦ { toFun := g.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ },
    map_add' := ⋯, map_smul' := ⋯ }
⊢ ∀ (ib : Set (ContinuousMultilinearMap 𝕜 E F) × Set ((i : ι) → E i) × Set G),
    IsVonNBounded 𝕜 ib.1 ∧ IsVonNBounded 𝕜 ib.2.1 ∧ ib.2.2 ∈ 𝓝 0 →
      ∃ ia,
        (IsVonNBounded 𝕜 ia.1 ∧ ia.2 ∈ 𝓝 0) ∧
          ∀ x ∈ {f | ∀ x ∈ ia.1, f x ∈ ia.2}, aux x ∈ {f | ∀ x ∈ ib.1, f x ∈ {f | MapsTo (⇑f) ib.2.1 ib.2.2}}
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: α : Type u_1
γ : Type u_3
inst✝ : TopologicalSpace γ
β : Type u_4
f : α → β
mα : MeasurableSpace α
μa : Measure α
mβ : MeasurableSpace β
μb : Measure β
hf : MeasurePreserving f μa μb
h₂ : MeasurableEmbedding f
g : β → γ
⊢ AEStronglyMeasurable (g ∘ f) μa ↔ AEStronglyMeasurable (g ∘ f) μa

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: case mp.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, (fun x ↦ ↑(f x) • g x) x = (fun x ↦ ↑(f x) • g' x) x

Tactic:
]

Goal state after:
case mp.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, (fun x ↦ ↑(f x) • g x) x = (fun x ↦ ↑(f x) • g' x) x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: case mp.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ, x ∈ {x | f x ≠ 0} → (fun x ↦ ↑(f x) • g x) x = (fun x ↦ ↑(f x) • g' x) x

Tactic:
]

Goal state after:
case mp.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ, x ∈ {x | f x ≠ 0} → (fun x ↦ ↑(f x) • g x) x = (fun x ↦ ↑(f x) • g' x) x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f x ≠ 0}
a : α
ha : ↑(f a) ≠ 0 → g a = g' a
h'a : f a ≠ 0
this : ↑(f a) ≠ 0
⊢ ↑(f a) • g' a = ↑(f a) • g' a

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: case mpr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
⊢ ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = (↑(f x))⁻¹ • g' x

Tactic:
]

Goal state after:
case mpr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
⊢ ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = (↑(f x))⁻¹ • g' x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
x : α
hx : ↑(f x) • g x = g' x
h'x : ↑(f x) ≠ 0
⊢ g x = g x

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
x : α
hx : ↑(f x) • g x = g' x
h'x : ↑(f x) ≠ 0
⊢ ↑(f x) ≠ 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
x : α
hx : ↑(f x) • g x = g' x
h'x : ↑(f x) ≠ 0
⊢ ↑(f x) ≠ 0
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : TopologicalSpace M
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p q : Submodule R M
h : IsCompl p q
⊢ Continuous[inst✝², inst✝²] ⇑(p.projection q h) ↔ Continuous[inst✝², inst✝²] (Subtype.val ∘ ⇑(p.projectionOnto q h))

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : TopologicalSpace M
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p q : Submodule R M
h : IsCompl p q
⊢ Continuous[inst✝², inst✝²] ⇑(p.projection q h) ↔ Continuous[inst✝², inst✝²] (Subtype.val ∘ ⇑(p.projectionOnto q h))
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : ContinuousSub M
h : IsTopCompl p q
⊢ Continuous[inst✝³, inst✝³] ⇑(LinearMap.id - p.projection q ⋯)

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : ContinuousSub M
h : IsTopCompl p q
⊢ Continuous[inst✝³, inst✝³] ⇑(LinearMap.id - p.projection q ⋯)
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : ContinuousSub M
h : IsTopCompl p q
x : M
⊢ x = x

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : T1Space ↥p
h : IsTopCompl p q
⊢ IsClosed[inst✝³] ↑(↑(p.projectionOntoL q h)).ker

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : T1Space ↥p
h : IsTopCompl p q
⊢ IsClosed[inst✝³] ↑(↑(p.projectionOntoL q h)).ker
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑q
⊢ IsClosed[inst✝³] (⇑(p.projectionOntoL q h) ⁻¹' {0})

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑q
⊢ IsClosed[inst✝³] (⇑(p.projectionOntoL q h) ⁻¹' {0})
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑(↑(p.projectionOntoL q h)).ker
⊢ IsClosed[inst✝³] (⇑(p.projectionOntoL q h) ⁻¹' {0})

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑(↑(p.projectionOntoL q h)).ker
⊢ IsClosed[inst✝³] (⇑(p.projectionOntoL q h) ⁻¹' {0})
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑q
this✝ : IsClosed[instTopologicalSpaceSubtype] {0}
this : T1Space ↥p
⊢ T0Space ↥p

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑q
this✝ : IsClosed[instTopologicalSpaceSubtype] {0}
this : T1Space ↥p
⊢ T0Space ↥p
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsCompl p q
⊢ Continuous[inst✝³, instTopologicalSpaceProd] ⇑(p.prodEquivOfIsCompl q ?m.64).symm ↔
    Continuous[inst✝³, instTopologicalSpaceProd] ⇑(p.prodEquivOfIsCompl q h).symm

pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsCompl p q
⊢ Continuous[instTopologicalSpaceProd, inst✝³] ⇑(p.prodEquivOfIsCompl q h)

pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsCompl p q
⊢ IsCompl p q

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsCompl p q
⊢ Continuous[instTopologicalSpaceProd, inst✝³] ⇑(p.prodEquivOfIsCompl q h)
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsCompl p q
⊢ IsTopCompl q p ↔ Continuous[inst✝³, instTopologicalSpaceSubtype] (⇑(p.quotientEquivOfIsCompl q h) ∘ ⇑p.mkQL)

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsCompl p q
⊢ IsTopCompl q p ↔ Continuous[inst✝³, instTopologicalSpaceSubtype] (⇑(p.quotientEquivOfIsCompl q h) ∘ ⇑p.mkQL)
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
h : μ s ≠ ∞
⊢ μ s = 0 ∨ μ s = ∞ ↔ μ s = 0

Tactic:
]

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
h : μ s ≠ ∞
⊢ μ s = 0 ∨ μ s = ∞ ↔ μ s = 0
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : s₁ ⊆ s₂
h₂ : μ s₂ = 0
h'₂ : μ s₂ ≠ ∞
⊢ μ.real s₁ = 0

Tactic:
]

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : s₁ ⊆ s₂
h₂ : μ s₂ = 0
h'₂ : μ s₂ ≠ ∞
⊢ μ.real s₁ = 0
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
h : μ s ≠ ∞
⊢ μ.real t ≤ μ.real (t ∪ s)

Tactic:
]

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
h : μ s ≠ ∞
⊢ μ.real t ≤ μ.real (t ∪ s)
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ (μ (s ∩ t) + μ (s \ t)).toReal = (μ s).toReal

pp: case ha
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ μ (s ∩ t) ≠ ∞

pp: case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ μ (s \ t) ≠ ∞

Tactic:
measure_inter_add_sdiff₀ s ht

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ (μ s).toReal = (μ s).toReal

case ha
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ μ (s ∩ t) ≠ ∞

case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ μ (s \ t) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ (μ s).toReal = (μ s).toReal

pp: case ha
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ μ (s ∩ t) ≠ ∞

pp: case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ μ (s \ t) ≠ ∞

Tactic:
]

Goal state after:
case ha
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ μ (s ∩ t) ≠ ∞

case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h : μ s ≠ ∞
⊢ μ (s \ t) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
this : μ (s ∪ t) ≠ ∞
⊢ μ.real t + μ.real (s \ t) + μ.real (s ∩ t) = μ.real (s ∩ t) + μ.real (s \ t) + μ.real t

Tactic:
]

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
this : μ (s ∪ t) ≠ ∞
⊢ μ.real t + μ.real (s \ t) + μ.real (s ∩ t) = μ.real (s ∩ t) + μ.real (s \ t) + μ.real t
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ.real s + μ.real t = μ.real s + μ.real t

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : NullMeasurableSet t μ
hd : AEDisjoint μ s t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ (μ s).toReal + (μ t).toReal = (μ s).toReal + (μ t).toReal

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
hd : AEDisjoint μ s t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ.real s + μ.real t = μ.real s + μ.real t

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsFiniteMeasure μ
hs : NullMeasurableSet s μ
⊢ μ.real univ = μ.real univ

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : MeasurableSet t
h : μ s ≠ ∞
⊢ (μ s).toReal = (μ s).toReal

pp: case ha
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : MeasurableSet t
h : μ s ≠ ∞
⊢ μ (s ∩ t) ≠ ∞

pp: case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : MeasurableSet t
h : μ s ≠ ∞
⊢ μ (s \ t) ≠ ∞

Tactic:
]

Goal state after:
case ha
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : MeasurableSet t
h : μ s ≠ ∞
⊢ μ (s ∩ t) ≠ ∞

case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
ht : MeasurableSet t
h : μ s ≠ ∞
⊢ μ (s \ t) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ (μ (s \ t) + μ (t \ s)).toReal = (μ (s \ t) + μ (t \ s)).toReal

pp: case ha
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ (s \ t) ≠ ∞

pp: case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ (t \ s) ≠ ∞

Tactic:
]

Goal state after:
case ha
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ (s \ t) ≠ ∞

case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ (t \ s) ≠ ∞