Theorem: VitaliFamily.ae_tendsto_div
File: Mathlib/MeasureTheory/Covering/Differentiation.lean

tactic: obtain ⟨w, w_count, w_dense, _, w_top⟩ :
  ∃ w : Set ℝ≥0∞, w.Countable ∧ Dense w ∧ 0 ∉ w ∧ ∞ ∉ w :=
  ENNReal.exists_countable_dense_no_zero_top
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)

tactic: have I : ∀ x ∈ w, x ≠ ∞ := fun x xs hx => w_top (hx ▸ xs)
state_before: case intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)

tactic: have B : ∀ᵐ x ∂μ, ∀ c ∈ w, ∀ d ∈ w, c < d →
    ¬((∃ᶠ a in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ a in v.filterAt x, d < ρ a / μ a) := by
  simpa only [ae_ball_iff w_count, ae_all_iff]
state_before: case intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
A :
  ∀ (c : ℝ≥0∞),
    c ∈ w →
      ∀ (d : ℝ≥0∞),
        d ∈ w →
          c < d →
            ∀ᵐ (x : α) ∂μ,
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)
⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)

tactic: filter_upwards [B]
state_before: case intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
A :
  ∀ (c : ℝ≥0∞),
    c ∈ w →
      ∀ (d : ℝ≥0∞),
        d ∈ w →
          c < d →
            ∀ᵐ (x : α) ∂μ,
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ (c : ℝ≥0∞),
      c ∈ w →
        ∀ (d : ℝ≥0∞),
          d ∈ w →
            c < d →
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)
⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)

tactic: intro x hx
state_before: case h
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
A :
  ∀ (c : ℝ≥0∞),
    c ∈ w →
      ∀ (d : ℝ≥0∞),
        d ∈ w →
          c < d →
            ∀ᵐ (x : α) ∂μ,
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ (c : ℝ≥0∞),
      c ∈ w →
        ∀ (d : ℝ≥0∞),
          d ∈ w →
            c < d →
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)
⊢ ∀ (a : α),
    (∀ (c : ℝ≥0∞),
        c ∈ w →
          ∀ (d : ℝ≥0∞),
            d ∈ w →
              c < d →
                ¬((∃ᶠ (a : Set α) in filterAt v a, ↑↑ρ a / ↑↑μ a < c) ∧
                    ∃ᶠ (a : Set α) in filterAt v a, d < ↑↑ρ a / ↑↑μ a)) →
      ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 c)

tactic: exact tendsto_of_no_upcrossings w_dense hx
state_before: case h
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
A :
  ∀ (c : ℝ≥0∞),
    c ∈ w →
      ∀ (d : ℝ≥0∞),
        d ∈ w →
          c < d →
            ∀ᵐ (x : α) ∂μ,
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ (c : ℝ≥0∞),
      c ∈ w →
        ∀ (d : ℝ≥0∞),
          d ∈ w →
            c < d →
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)
x : α
hx :
  ∀ (c : ℝ≥0∞),
    c ∈ w →
      ∀ (d : ℝ≥0∞),
        d ∈ w →
          c < d →
            ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)
⊢ ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)

tactic: intro c hc d hd hcd
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
⊢ ∀ (c : ℝ≥0∞),
    c ∈ w →
      ∀ (d : ℝ≥0∞),
        d ∈ w →
          c < d →
            ∀ᵐ (x : α) ∂μ,
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)

tactic: lift c to ℝ≥0 using I c hc
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0∞
hc : c ∈ w
d : ℝ≥0∞
hd : d ∈ w
hcd : c < d
⊢ ∀ᵐ (x : α) ∂μ,
    ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)

tactic: lift d to ℝ≥0 using I d hd
state_before: case intro
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
d : ℝ≥0∞
hd : d ∈ w
c : ℝ≥0
hc : ↑c ∈ w
hcd : ↑c < d
⊢ ∀ᵐ (x : α) ∂μ,
    ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)

tactic: apply v.null_of_frequently_le_of_frequently_ge hρ (ENNReal.coe_lt_coe.1 hcd)
state_before: case intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
⊢ ∀ᵐ (x : α) ∂μ,
    ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, ↑d < ↑↑ρ a / ↑↑μ a)

tactic: simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually,
  mem_setOf_eq, mem_compl_iff, not_forall]
state_before: case intro.intro.hc
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
⊢ ∀ (x : α),
    x ∈
        {x |
            (fun x =>
                ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧
                    ∃ᶠ (a : Set α) in filterAt v x, ↑d < ↑↑ρ a / ↑↑μ a))
              x}ᶜ →
      ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a

tactic: intro x h1x _
state_before: case intro.intro.hc
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
⊢ ∀ (x : α),
    (∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) →
      (∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x) → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a

tactic: apply h1x.mono fun a ha => ?_
state_before: case intro.intro.hc
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
h1x : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c
a✝ : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x
⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a

tactic: refine' (ENNReal.div_le_iff_le_mul _ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
h1x : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c
a✝ : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x
a : Set α
ha : ↑↑ρ a / ↑↑μ a < ↑c
⊢ ↑↑ρ a ≤ ↑c * ↑↑μ a

tactic: simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff]
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
h1x : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c
a✝ : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x
a : Set α
ha : ↑↑ρ a / ↑↑μ a < ↑c
⊢ ↑↑μ a ≠ 0 ∨ ↑c ≠ ⊤

tactic: simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually,
  mem_setOf_eq, mem_compl_iff, not_forall]
state_before: case intro.intro.hd
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
⊢ ∀ (x : α),
    x ∈
        {x |
            (fun x =>
                ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧
                    ∃ᶠ (a : Set α) in filterAt v x, ↑d < ↑↑ρ a / ↑↑μ a))
              x}ᶜ →
      ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a

tactic: intro x _ h2x
state_before: case intro.intro.hd
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
⊢ ∀ (x : α),
    (∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) →
      (∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x) → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a

tactic: apply h2x.mono fun a ha => ?_
state_before: case intro.intro.hd
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
a✝ : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c
h2x : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x
⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a

tactic: exact ENNReal.mul_le_of_le_div ha.le
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
a✝ : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c
h2x : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x
a : Set α
ha : ↑d < ↑↑ρ a / ↑↑μ a
⊢ ↑d * ↑↑μ a ≤ ↑↑ρ a

tactic: simpa only [ae_ball_iff w_count, ae_all_iff]
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.2032110
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : Set.Countable w
w_dense : Dense w
left✝ : ¬0 ∈ w
w_top : ¬⊤ ∈ w
I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤
A :
  ∀ (c : ℝ≥0∞),
    c ∈ w →
      ∀ (d : ℝ≥0∞),
        d ∈ w →
          c < d →
            ∀ᵐ (x : α) ∂μ,
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)
⊢ ∀ᵐ (x : α) ∂μ,
    ∀ (c : ℝ≥0∞),
      c ∈ w →
        ∀ (d : ℝ≥0∞),
          d ∈ w →
            c < d →
              ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a)