Theorem: ENNReal.lintegral_Lp_add_le_aux
File: Mathlib/MeasureTheory/Integral/MeanInequalities.lean

tactic: have hp_not_nonpos : ¬p ≤ 0 := by simp [hpq.pos]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

tactic: have htop_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
  by_contra h
  exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

tactic: have h0_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0 := by
  simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

tactic: suffices h :
  1 ≤
    (∫⁻ a : α, (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a : α, g a ^ p ∂μ) ^ (1 / p))
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

tactic: have h :
  (∫⁻ a : α, (f + g) a ^ p ∂μ) ≤
    ((∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a : α, g a ^ p ∂μ) ^ (1 / p)) *
      (∫⁻ a : α, (f + g) a ^ p ∂μ) ^ (1 / q) :=
  lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add hpq hf hf_top hg hg_top
state_before: case h
α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))

tactic: have h_one_div_q : 1 / q = 1 - 1 / p := by
  nth_rw 2 [← hpq.inv_add_inv_conj]
  ring
state_before: case h
α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) * (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / q)
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))

tactic: simp_rw [h_one_div_q, sub_eq_add_neg 1 (1 / p), ENNReal.rpow_add _ _ h_add_zero h_add_top,
  rpow_one] at h
state_before: case h
α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) * (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / q)
h_one_div_q : 1 / q = 1 - 1 / p
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))

tactic: conv_rhs at h => enter [2]; rw [mul_comm]
state_before: case h
α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h_one_div_q : 1 / q = 1 - 1 / p
h :
  (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) *
      ((∫⁻ (a : α), (f + g) a ^ p ∂μ) * (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)))
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))

tactic: conv_lhs at h => rw [← one_mul (∫⁻ a : α, (f + g) a ^ p ∂μ)]
state_before: case h
α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h_one_div_q : 1 / q = 1 - 1 / p
h :
  (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) *
      ((∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) * ∫⁻ (a : α), (f + g) a ^ p ∂μ)
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))

tactic: rwa [← mul_assoc, ENNReal.mul_le_mul_right h_add_zero h_add_top, mul_comm] at h
state_before: case h
α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h_one_div_q : 1 / q = 1 - 1 / p
h :
  (1 * ∫⁻ (a : α), (f + g) a ^ p ∂μ) ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) *
      ((∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) * ∫⁻ (a : α), (f + g) a ^ p ∂μ)
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))

tactic: simp [hpq.pos]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
⊢ ¬p ≤ 0

tactic: by_contra h
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤

tactic: exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
h : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) = ⊤
⊢ False

tactic: simp [hpq.nonneg]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
h : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) = ⊤
⊢ 0 ≤ 1 / p

tactic: simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0

tactic: rwa [← mul_le_mul_left h0_rpow htop_rpow, ← mul_assoc, ← rpow_add _ _ h_add_zero h_add_top, ←
  sub_eq_add_neg, _root_.sub_self, rpow_zero, one_mul, mul_one] at h
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

tactic: nth_rw 2 [← hpq.inv_add_inv_conj]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) * (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / q)
⊢ 1 / q = 1 - 1 / p

tactic: ring
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : MeasureTheory.Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤
hg : AEMeasurable g
hg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤
h_add_zero : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ 0
h_add_top : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) * (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / q)
⊢ 1 / q = 1 / p + 1 / q - 1 / p