-- Mathlib/MeasureTheory/Integral/SetToL1.lean
theorem setToSimpleFunc_mono {T : Set α → G' →L[ℝ] G''} (h_add : FinMeasAdditive μ T)
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →ₛ G'}
    (hfi : Integrable f μ) (hgi : Integrable g μ) (hfg : f ≤ g) :
    setToSimpleFunc T f ≤ setToSimpleFunc T g

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
x : α
⊢ f x ≤ g x

Tactic:
exact hfg x

Result:
no goals

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
⊢ setToSimpleFunc T f ≤ setToSimpleFunc T g

Tactic:
rw [← sub_nonneg, ← setToSimpleFunc_sub T h_add hgi hfi]

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
⊢ 0 ≤ setToSimpleFunc T (g - f)

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
⊢ 0 ≤ g - f

Tactic:
intro x

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
x : α
⊢ 0 x ≤ (g - f) x

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
x : α
⊢ 0 x ≤ (g - f) x

Tactic:
simp only [coe_sub, sub_nonneg, coe_zero, Pi.zero_apply, Pi.sub_apply]

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
x : α
⊢ f x ≤ g x

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
⊢ 0 ≤ setToSimpleFunc T (g - f)

Tactic:
refine setToSimpleFunc_nonneg' T hT_nonneg _ ?_ (hgi.sub hfi)

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
⊢ 0 ≤ g - f