Theorem: ENNReal.tsum_mul_left
File: Mathlib/Topology/Instances/ENNReal.lean

tactic: by_cases hf : ∀ i, f i = 0
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
⊢ ∑' (i : α), a * f i = a * ∑' (i : α), f i
state_after: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ∀ (i : α), f i = 0
⊢ ∑' (i : α), a * f i = a * ∑' (i : α), f i

case neg
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ¬∀ (i : α), f i = 0
⊢ ∑' (i : α), a * f i = a * ∑' (i : α), f i

tactic: simp [hf]
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ∀ (i : α), f i = 0
⊢ ∑' (i : α), a * f i = a * ∑' (i : α), f i
state_after: no goals

tactic: rw [← ENNReal.tsum_eq_zero] at hf
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ¬∀ (i : α), f i = 0
⊢ ∑' (i : α), a * f i = a * ∑' (i : α), f i
state_after: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ¬∑' (i : α), f i = 0
⊢ ∑' (i : α), a * f i = a * ∑' (i : α), f i

tactic: have : Tendsto (fun s : Finset α => ∑ j ∈ s, a * f j) atTop (𝓝 (a * ∑' i, f i)) := by
  simp only [← Finset.mul_sum]
  exact ENNReal.Tendsto.const_mul ENNReal.summable.hasSum (Or.inl hf)
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ¬∑' (i : α), f i = 0
⊢ ∑' (i : α), a * f i = a * ∑' (i : α), f i
state_after: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ¬∑' (i : α), f i = 0
this : Tendsto (fun s => ∑ j ∈ s, a * f j) atTop (𝓝 (a * ∑' (i : α), f i))
⊢ ∑' (i : α), a * f i = a * ∑' (i : α), f i

tactic: exact HasSum.tsum_eq this
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ¬∑' (i : α), f i = 0
this : Tendsto (fun s => ∑ j ∈ s, a * f j) atTop (𝓝 (a * ∑' (i : α), f i))
⊢ ∑' (i : α), a * f i = a * ∑' (i : α), f i
state_after: no goals

tactic: simp only [← Finset.mul_sum]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ¬∑' (i : α), f i = 0
⊢ Tendsto (fun s => ∑ j ∈ s, a * f j) atTop (𝓝 (a * ∑' (i : α), f i))
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ¬∑' (i : α), f i = 0
⊢ Tendsto (fun s => a * ∑ i ∈ s, f i) atTop (𝓝 (a * ∑' (i : α), f i))

tactic: exact ENNReal.Tendsto.const_mul ENNReal.summable.hasSum (Or.inl hf)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f g : α → ℝ≥0∞
hf : ¬∑' (i : α), f i = 0
⊢ Tendsto (fun s => a * ∑ i ∈ s, f i) atTop (𝓝 (a * ∑' (i : α), f i))
state_after: no goals