Theorem: MeasureTheory.snorm_add_le
File: Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean

tactic: by_cases hp0 : p = 0
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : p = 0
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ

case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ

tactic: by_cases hp_top : p = ∞
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : p = ⊤
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ

case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ⊤
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ

tactic: have hp1_real : 1 ≤ p.toReal := by
  rwa [← ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top hp_top]
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ⊤
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ⊤
hp1_real : 1 ≤ p.toReal
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ

tactic: repeat rw [snorm_eq_snorm' hp0 hp_top]
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ⊤
hp1_real : 1 ≤ p.toReal
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ⊤
hp1_real : 1 ≤ p.toReal
⊢ snorm' (f + g) p.toReal μ ≤ snorm' f p.toReal μ + snorm' g p.toReal μ

tactic: exact snorm'_add_le hf hg hp1_real
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ⊤
hp1_real : 1 ≤ p.toReal
⊢ snorm' (f + g) p.toReal μ ≤ snorm' f p.toReal μ + snorm' g p.toReal μ
state_after: no goals

tactic: simp [hp0]
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : p = 0
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ
state_after: no goals

tactic: simp [hp_top, snormEssSup_add_le]
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : p = ⊤
⊢ snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ
state_after: no goals

tactic: rwa [← ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top hp_top]
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ⊤
⊢ 1 ≤ p.toReal
state_after: no goals

tactic: rw [snorm_eq_snorm' hp0 hp_top]
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ⊤
hp1_real : 1 ≤ p.toReal
⊢ snorm' (f + g) p.toReal μ ≤ snorm' f p.toReal μ + snorm g p μ
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g✝ f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ⊤
hp1_real : 1 ≤ p.toReal
⊢ snorm' (f + g) p.toReal μ ≤ snorm' f p.toReal μ + snorm' g p.toReal μ