-- 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.toReal μ < ⊤

Tactic:
rw [eLpNorm_eq_eLpNorm' hq0 hq_top] at hfq_lt_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.toReal 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 < 0
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
hp_eq_zero : p = 0
⊢ False

Tactic:
exact (lt_irrefl _) hp_pos

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
hp_top : p = ⊤
hq_top : q = ⊤
⊢ eLpNorm f ⊤ μ < ⊤

Tactic:
rwa [hq_top] at hfq_lt_top

Result:
no goals

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
hp_eq_zero : p = 0
⊢ False

Tactic:
rw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos

Result:
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 < 0
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
hp_eq_zero : p = 0
⊢ False

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 = ⊤
⊢ eLpNormEssSup f μ * μ Set.univ ^ (1 / p.toReal) < ⊤

Tactic:
refine ENNReal.mul_lt_top hfq_lt_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 : eLpNormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ μ Set.univ ^ (1 / p.toReal) < ⊤