Theorem: exists_mul_eq_setInterval_Iic (ForML/IntegralMeanValue.lean)

State before:
a : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf_cont : ContinuousOn f (Set.Iic a)
hg_int : IntegrableOn g (Set.Iic a)
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Iic a)] g
h_int : IntegrableOn (fun x => f x * g x) (Set.Iic a)
⊢ ∃ c ∈ Set.Iic a, ∫ (x : ℝ) in Set.Iic a, f x * g x ∂μ = f c * ∫ (x : ℝ) in Set.Iic a, g x ∂μ
Tactic:
apply exists_mul_eq_setInterval hf_cont hg_int hg_nonneg h_int <;> try simp
State after:
a : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf_cont : ContinuousOn f (Set.Iic a)
hg_int : IntegrableOn g (Set.Iic a)
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Iic a)] g
h_int : IntegrableOn (fun x => f x * g x) (Set.Iic a)
⊢ IsPreconnected (Set.Iic a)

a : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf_cont : ContinuousOn f (Set.Iic a)
hg_int : IntegrableOn g (Set.Iic a)
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Iic a)] g
h_int : IntegrableOn (fun x => f x * g x) (Set.Iic a)
⊢ IsClosed (Set.Iic a)

State before:
a : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf_cont : ContinuousOn f (Set.Iic a)
hg_int : IntegrableOn g (Set.Iic a)
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Iic a)] g
h_int : IntegrableOn (fun x => f x * g x) (Set.Iic a)
⊢ IsPreconnected (Set.Iic a)

a : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf_cont : ContinuousOn f (Set.Iic a)
hg_int : IntegrableOn g (Set.Iic a)
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Iic a)] g
h_int : IntegrableOn (fun x => f x * g x) (Set.Iic a)
⊢ IsClosed (Set.Iic a)
Tactic:
repeat simp [isClosed_Iic, isPreconnected_Iic]
State after:
no goals

State before:
a : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf_cont : ContinuousOn f (Set.Iic a)
hg_int : IntegrableOn g (Set.Iic a)
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Iic a)] g
h_int : IntegrableOn (fun x => f x * g x) (Set.Iic a)
⊢ MeasurableSet (Set.Iic a)
Tactic:
simp
State after:
no goals

State before:
a : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf_cont : ContinuousOn f (Set.Iic a)
hg_int : IntegrableOn g (Set.Iic a)
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Iic a)] g
h_int : IntegrableOn (fun x => f x * g x) (Set.Iic a)
⊢ IsClosed (Set.Iic a)
Tactic:
simp [isClosed_Iic, isPreconnected_Iic]
State after:
no goals