Theorem: MeasureTheory.lintegral_const_mul'
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ
state_after: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : r = 0
⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ

case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ
tactic: by_cases h : r = 0
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
tactic: apply le_antisymm _ (lintegral_const_mul_le r f)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
tactic: have rinv : r * r⁻¹ = 1 := ENNReal.mul_inv_cancel h hr
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
rinv' : r⁻¹ * r = 1
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
tactic: have rinv' : r⁻¹ * r = 1 := by
    rw [mul_comm]
    exact rinv
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
rinv' : r⁻¹ * r = 1
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
rinv' : r⁻¹ * r = 1
this : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), r⁻¹ * (r * f a) ∂μ
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
tactic: have := lintegral_const_mul_le (μ := μ) r⁻¹ fun x => r * f x
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
rinv' : r⁻¹ * r = 1
this : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), r⁻¹ * (r * f a) ∂μ
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
rinv' : r⁻¹ * r = 1
this : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), f a ∂μ
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
tactic: simp? [(mul_assoc _ _ _).symm, rinv'] at this says
    simp only [(mul_assoc _ _ _).symm, rinv', one_mul] at this
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
rinv' : r⁻¹ * r = 1
this : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), f a ∂μ
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ
state_after: no goals
tactic: simpa [(mul_assoc _ _ _).symm, rinv] using mul_le_mul_left' this r