-- Mathlib/MeasureTheory/Integral/SetToL1.lean
theorem setToFun_mono {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C)
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α → G'}
    (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    setToFun μ T hT f ≤ setToFun μ T hT g

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁴ : NormedLatticeAddCommGroup G''
inst✝³ : NormedSpace ℝ G''
inst✝² : CompleteSpace G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α → G'
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᶠ[ae μ] g
⊢ 0 ≤ setToFun μ T hT (g - f)

Tactic:
refine setToFun_nonneg hT hT_nonneg (hfg.mono fun a ha => ?_)

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁴ : NormedLatticeAddCommGroup G''
inst✝³ : NormedSpace ℝ G''
inst✝² : CompleteSpace G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α → G'
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᶠ[ae μ] g
a : α
ha : f a ≤ g a
⊢ 0 a ≤ (g - f) a

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁴ : NormedLatticeAddCommGroup G''
inst✝³ : NormedSpace ℝ G''
inst✝² : CompleteSpace G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α → G'
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᶠ[ae μ] g
a : α
ha : f a ≤ g a
⊢ 0 a ≤ (g - f) a

Tactic:
rw [Pi.sub_apply, Pi.zero_apply, sub_nonneg]

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁴ : NormedLatticeAddCommGroup G''
inst✝³ : NormedSpace ℝ G''
inst✝² : CompleteSpace G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α → G'
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᶠ[ae μ] g
a : α
ha : f a ≤ g a
⊢ f a ≤ g a

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁴ : NormedLatticeAddCommGroup G''
inst✝³ : NormedSpace ℝ G''
inst✝² : CompleteSpace G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α → G'
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᶠ[ae μ] g
⊢ setToFun μ T hT f ≤ setToFun μ T hT g

Tactic:
rw [← sub_nonneg, ← setToFun_sub hT hg hf]

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁴ : NormedLatticeAddCommGroup G''
inst✝³ : NormedSpace ℝ G''
inst✝² : CompleteSpace G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α → G'
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᶠ[ae μ] g
⊢ 0 ≤ setToFun μ T hT (g - f)

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁴ : NormedLatticeAddCommGroup G''
inst✝³ : NormedSpace ℝ G''
inst✝² : CompleteSpace G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α → G'
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᶠ[ae μ] g
a : α
ha : f a ≤ g a
⊢ f a ≤ g a

Tactic:
exact ha

Result:
no goals