/-- Hölder's inequality for the integral of a product of norms. The integral of the product of two
norms of functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are
conjugate exponents. -/
theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
    (hpq : p.IsConjugateExponent q) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) := by
  -- translate the Bochner integrals into Lebesgue integrals.
  rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae,
    integral_eq_lintegral_of_nonneg_ae]
  rotate_left
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · exact (hg.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · exact (hf.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
  · exact eventually_of_forall fun x => mul_nonneg (norm_nonneg _) (norm_nonneg _)
  · exact hf.1.norm.mul hg.1.norm
  rw [ENNReal.toReal_rpow, ENNReal.toReal_rpow, ← ENNReal.toReal_mul]
  -- replace norms by nnnorm
  have h_left : ∫⁻ a, ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ =
      ∫⁻ a, ((fun x => (‖f x‖₊ : ℝ≥0∞)) * fun x => (‖g x‖₊ : ℝ≥0∞)) a ∂μ := by
    simp_rw [Pi.mul_apply, ← ofReal_norm_eq_coe_nnnorm, ENNReal.ofReal_mul (norm_nonneg _)]
  have h_right_f : ∫⁻ a, ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ p ∂μ := by
    refine' lintegral_congr fun x => _
    rw [← ofReal_norm_eq_coe_nnnorm, ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.nonneg]
  have h_right_g : ∫⁻ a, ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ a, (‖g a‖₊ : ℝ≥0∞) ^ q ∂μ := by
    refine' lintegral_congr fun x => _
    rw [← ofReal_norm_eq_coe_nnnorm, ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.symm.nonneg]
  rw [h_left, h_right_f, h_right_g]
  -- we can now apply `ENNReal.lintegral_mul_le_Lp_mul_Lq` (up to the `toReal` application)
  

Proof state:
α : Type u_1
E✝ : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
hE : CompleteSpace E✝
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E✝
inst✝⁸ : SMulCommClass ℝ 𝕜 E✝
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g✝ : α → E✝
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : Real.IsConjugateExponent p q
hf : Memℒp f (ENNReal.ofReal p)
hg : Memℒp g (ENNReal.ofReal q)
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x => ↑‖f x‖₊) * fun x => ↑‖g x‖₊) a ∂μ
h_right_f : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ^ p ∂μ
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ↑‖g a‖₊ ^ q ∂μ
⊢ ENNReal.toReal (∫⁻ (a : α), ((fun x => ↑‖f x‖₊) * fun x => ↑‖g x‖₊) a ∂μ) ≤
    ENNReal.toReal ((∫⁻ (a : α), ↑‖f a‖₊ ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ↑‖g a‖₊ ^ q ∂μ) ^ (1 / q))

Next tactic:
refine' ENNReal.toReal_mono _ _