Theorem: MeasureTheory.VectorMeasure.trim_apply
File: Mathlib/MeasureTheory/Measure/VectorMeasure.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
⊢ (fun i => if MeasurableSet i then ↑v i else 0) ∅ = 0
state_after: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
⊢ (if MeasurableSet ∅ then ↑v ∅ else 0) = 0
tactic: dsimp only
state_before: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
⊢ (if MeasurableSet ∅ then ↑v ∅ else 0) = 0
state_after: no goals
tactic: rw [if_pos (@MeasurableSet.empty _ m), v.empty]
state_before: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : ¬MeasurableSet i
⊢ (fun i => if MeasurableSet i then ↑v i else 0) i = 0
state_after: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : ¬MeasurableSet i
⊢ (if MeasurableSet i then ↑v i else 0) = 0
tactic: dsimp only
state_before: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : ¬MeasurableSet i
⊢ (if MeasurableSet i then ↑v i else 0) = 0
state_after: no goals
tactic: rw [if_neg hi]
state_before: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
⊢ HasSum (fun i => (fun i => if MeasurableSet i then ↑v i else 0) (f i))
    ((fun i => if MeasurableSet i then ↑v i else 0) (⋃ i, f i))
state_after: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
⊢ HasSum (fun i => if MeasurableSet (f i) then ↑v (f i) else 0) (if MeasurableSet (⋃ i, f i) then ↑v (⋃ i, f i) else 0)
tactic: dsimp only
state_before: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
⊢ HasSum (fun i => if MeasurableSet (f i) then ↑v (f i) else 0) (if MeasurableSet (⋃ i, f i) then ↑v (⋃ i, f i) else 0)
state_after: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
hf₁' : ∀ (k : ℕ), MeasurableSet (f k)
⊢ HasSum (fun i => if MeasurableSet (f i) then ↑v (f i) else 0) (if MeasurableSet (⋃ i, f i) then ↑v (⋃ i, f i) else 0)
tactic: have hf₁' : ∀ k, MeasurableSet[n] (f k) := fun k => hle _ (hf₁ k)
state_before: α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
hf₁' : ∀ (k : ℕ), MeasurableSet (f k)
⊢ HasSum (fun i => if MeasurableSet (f i) then ↑v (f i) else 0) (if MeasurableSet (⋃ i, f i) then ↑v (⋃ i, f i) else 0)
state_after: case h.e'_5
α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
hf₁' : ∀ (k : ℕ), MeasurableSet (f k)
⊢ (fun i => if MeasurableSet (f i) then ↑v (f i) else 0) = fun i => ↑v (f i)

case h.e'_6
α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
hf₁' : ∀ (k : ℕ), MeasurableSet (f k)
⊢ (if MeasurableSet (⋃ i, f i) then ↑v (⋃ i, f i) else 0) = ↑v (⋃ i, f i)
tactic: convert v.m_iUnion hf₁' hf₂ using 1