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

Proof steps:
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
state_after: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ∫⁻ (a : α), f a ^ p ∂μ = 0
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)

case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
tactic: by_cases hf_zero : ∫⁻ a, f a ^ p ∂μ = 0
state_before: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
state_after: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ∫⁻ (a : α), g a ^ q ∂μ = 0
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)

case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
tactic: by_cases hg_zero : ∫⁻ a, g a ^ q ∂μ = 0
state_before: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
state_after: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ = ⊤
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)

case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0
hf_top : ¬∫⁻ (a : α), f a ^ p ∂μ = ⊤
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
tactic: by_cases hf_top : ∫⁻ a, f a ^ p ∂μ = ⊤
state_before: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0
hf_top : ¬∫⁻ (a : α), f a ^ p ∂μ = ⊤
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
state_after: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0
hf_top : ¬∫⁻ (a : α), f a ^ p ∂μ = ⊤
hg_top : ∫⁻ (a : α), g a ^ q ∂μ = ⊤
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)

case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0
hf_top : ¬∫⁻ (a : α), f a ^ p ∂μ = ⊤
hg_top : ¬∫⁻ (a : α), g a ^ q ∂μ = ⊤
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
tactic: by_cases hg_top : ∫⁻ a, g a ^ q ∂μ = ⊤
state_before: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ✝ μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0
hf_top : ¬∫⁻ (a : α), f a ^ p ∂μ = ⊤
hg_top : ¬∫⁻ (a : α), g a ^ q ∂μ = ⊤
⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
state_after: no goals
tactic: exact ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top hpq hf hf_top hg_top hf_zero hg_zero