Theorem: MeasureTheory.eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ
File: Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean

Proof steps:
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
⊢ eLpNorm' f q μ ≤ eLpNormEssSup f μ * μ Set.univ ^ (1 / q)
state_after: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ eLpNorm' f q μ ≤ eLpNormEssSup f μ * μ Set.univ ^ (1 / q)
tactic: have h_le : (∫⁻ a : α, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) ≤ ∫⁻ _ : α, eLpNormEssSup f μ ^ q ∂μ := by
    refine lintegral_mono_ae ?_
    have h_nnnorm_le_eLpNorm_ess_sup := coe_nnnorm_ae_le_eLpNormEssSup f μ
    exact h_nnnorm_le_eLpNorm_ess_sup.mono fun x hx => by gcongr
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ eLpNorm' f q μ ≤ eLpNormEssSup f μ * μ Set.univ ^ (1 / q)
state_after: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ) ^ (1 / q) ≤ eLpNormEssSup f μ ^ 1 * μ Set.univ ^ (1 / q)
tactic: rw [eLpNorm', ← ENNReal.rpow_one (eLpNormEssSup f μ)]
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ) ^ (1 / q) ≤ eLpNormEssSup f μ ^ 1 * μ Set.univ ^ (1 / q)
state_after: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ) ^ (1 / q) ≤ eLpNormEssSup f μ ^ (q * q⁻¹) * μ Set.univ ^ (1 / q)
tactic: nth_rw 2 [← mul_inv_cancel (ne_of_lt hq_pos).symm]
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ) ^ (1 / q) ≤ eLpNormEssSup f μ ^ (q * q⁻¹) * μ Set.univ ^ (1 / q)
state_after: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ) ^ q⁻¹ ≤ (eLpNormEssSup f μ ^ q * μ Set.univ) ^ q⁻¹
tactic: rw [ENNReal.rpow_mul, one_div, ← ENNReal.mul_rpow_of_nonneg _ _ (by simp [hq_pos.le] : 0 ≤ q⁻¹)]
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ) ^ q⁻¹ ≤ (eLpNormEssSup f μ ^ q * μ Set.univ) ^ q⁻¹
state_after: case h₁
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ eLpNormEssSup f μ ^ q * μ Set.univ
tactic: gcongr
state_before: case h₁
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : α → E
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ ≤ eLpNormEssSup f μ ^ q * μ Set.univ
state_after: no goals
tactic: rwa [lintegral_const] at h_le