Theorem: blimsup_thickening_mul_ae_eq_aux
File: Mathlib/MeasureTheory/Covering/LiminfLimsup.lean

Proof steps:
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
tactic: have h₁ := blimsup_cthickening_ae_eq_blimsup_thickening (s := s) μ hr hr'
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
tactic: have h₂ := blimsup_cthickening_mul_ae_eq μ p s hM r hr
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
tactic: replace hr : Tendsto (fun i => M * r i) atTop (𝓝 0) := by convert hr.const_mul M; simp
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
tactic: replace hr' : ∀ᶠ i in atTop, p i → 0 < M * r i := hr'.mono fun i hi hip ↦ mul_pos hM (hi hip)
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i
h₃ :
  blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (M * r i) (s i)) atTop p
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
tactic: have h₃ := blimsup_cthickening_ae_eq_blimsup_thickening (s := s) μ hr hr'
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
hr : Tendsto (fun i => M * r i) atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i
h₃ :
  blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (M * r i) (s i)) atTop p
⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p
state_after: no goals
tactic: exact h₃.symm.trans (h₂.trans h₁)