-- Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean
theorem MemLp.mono_exponent {p q : ℝ≥0∞} [IsFiniteMeasure μ] {f : α → E} (hfq : MemLp f q μ)
    (hpq : p ≤ q) : MemLp f p μ

Proof state:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
hpq_real : p.toReal ≤ q.toReal
⊢ eLpNorm f p μ < ⊤

Tactic:
rw [eLpNorm_eq_eLpNorm' hp0 hp_top]

Result:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
hpq_real : p.toReal ≤ q.toReal
⊢ eLpNorm' f p.toReal μ < ⊤

Proof state:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
⊢ p ≤ 0

Tactic:
rwa [hq_eq_zero] at hpq

Result:
no goals

Proof state:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
⊢ eLpNorm f p μ < ⊤

Tactic:
have hpq_real : p.toReal ≤ q.toReal := ENNReal.toReal_mono hq_top hpq

Result:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
hpq_real : p.toReal ≤ q.toReal
⊢ eLpNorm f p μ < ⊤

Proof state:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
⊢ MemLp f p μ

Tactic:
refine ⟨hfq_m, ?_⟩

Result:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
⊢ eLpNorm f p μ < ⊤

Proof state:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ 0 ≤ 1 / p.toReal

Tactic:
simp [hp_pos.le]

Result:
no goals

Proof state:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p = 0
⊢ MemLp f p μ

Tactic:
rwa [hp0, memLp_zero_iff_aestronglyMeasurable]

Result:
no goals

Proof state:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ μ Set.univ ^ (1 / p.toReal) < ⊤

Tactic:
exact ENNReal.rpow_lt_top_of_nonneg (by simp [hp_pos.le]) (measure_ne_top μ Set.univ)

Result:
no goals

Proof state:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ eLpNorm' f p.toReal μ < ⊤

Tactic:
refine lt_of_le_of_lt (eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ hp_pos) ?_

Result:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ eLpNormEssSup f μ * μ Set.univ ^ (1 / p.toReal) < ⊤

Proof state:
case intro
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
⊢ MemLp f p μ

Tactic:
by_cases hp0 : p = 0

Result:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p = 0
⊢ MemLp f p μ

case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : ¬p = 0
⊢ MemLp f p μ

Proof state:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
⊢ eLpNorm f p μ < ⊤

Tactic:
have hq0 : q ≠ 0 := by
  by_contra hq_eq_zero
  have hp_eq_zero : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq) (zero_le _)
  rw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos
  exact (lt_irrefl _) hp_pos

Result:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
⊢ eLpNorm f p μ < ⊤

Proof state:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : p = ⊤
hq_top : q = ⊤
⊢ eLpNorm f p μ < ⊤

Tactic:
rw [hp_top]

Result:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : p = ⊤
hq_top : q = ⊤
⊢ eLpNorm f ⊤ μ < ⊤

Proof state:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : ¬p = 0
⊢ MemLp f p μ

Tactic:
rw [← Ne] at hp0

Result:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ⊤
hp0 : p ≠ 0
⊢ MemLp f p μ