Theorem: PartitionOfUnity.sum_finsupport_smul_eq_finsum
File: Mathlib/Topology/PartitionOfUnity.lean

Proof steps:
state_before: ι : Type u
X : Type v
inst✝⁶ : TopologicalSpace X
E : Type u_1
inst✝⁵ : AddCommMonoid E
inst✝⁴ : SMulWithZero ℝ E
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul ℝ E
s✝ : Set X
f : PartitionOfUnity ι X s✝
s : Set X
ρ : PartitionOfUnity ι X s
x₀✝ x₀ : X
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module ℝ M
φ : ι → X → M
⊢ ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀ • φ i x₀ = ∑ᶠ (i : ι), (ρ i) x₀ • φ i x₀
state_after: ι : Type u
X : Type v
inst✝⁶ : TopologicalSpace X
E : Type u_1
inst✝⁵ : AddCommMonoid E
inst✝⁴ : SMulWithZero ℝ E
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul ℝ E
s✝ : Set X
f : PartitionOfUnity ι X s✝
s : Set X
ρ : PartitionOfUnity ι X s
x₀✝ x₀ : X
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module ℝ M
φ : ι → X → M
⊢ (support fun i => (ρ i) x₀ • φ i x₀) ⊆ ↑(ρ.finsupport x₀)
tactic: apply (finsum_eq_sum_of_support_subset _ _).symm
state_before: ι : Type u
X : Type v
inst✝⁶ : TopologicalSpace X
E : Type u_1
inst✝⁵ : AddCommMonoid E
inst✝⁴ : SMulWithZero ℝ E
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul ℝ E
s✝ : Set X
f : PartitionOfUnity ι X s✝
s : Set X
ρ : PartitionOfUnity ι X s
x₀✝ x₀ : X
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module ℝ M
φ : ι → X → M
⊢ (support fun i => (ρ i) x₀ • φ i x₀) ⊆ ↑(ρ.finsupport x₀)
state_after: ι : Type u
X : Type v
inst✝⁶ : TopologicalSpace X
E : Type u_1
inst✝⁵ : AddCommMonoid E
inst✝⁴ : SMulWithZero ℝ E
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul ℝ E
s✝ : Set X
f : PartitionOfUnity ι X s✝
s : Set X
ρ : PartitionOfUnity ι X s
x₀✝ x₀ : X
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module ℝ M
φ : ι → X → M
this : (fun i => (ρ i) x₀ • φ i x₀) = (fun i => (ρ i) x₀) • fun i => φ i x₀
⊢ (support fun i => (ρ i) x₀ • φ i x₀) ⊆ ↑(ρ.finsupport x₀)
tactic: have : (fun i ↦ (ρ i) x₀ • φ i x₀) = (fun i ↦ (ρ i) x₀) • (fun i ↦ φ i x₀) :=
    funext fun _ => (Pi.smul_apply' _ _ _).symm
state_before: ι : Type u
X : Type v
inst✝⁶ : TopologicalSpace X
E : Type u_1
inst✝⁵ : AddCommMonoid E
inst✝⁴ : SMulWithZero ℝ E
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul ℝ E
s✝ : Set X
f : PartitionOfUnity ι X s✝
s : Set X
ρ : PartitionOfUnity ι X s
x₀✝ x₀ : X
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module ℝ M
φ : ι → X → M
this : (fun i => (ρ i) x₀ • φ i x₀) = (fun i => (ρ i) x₀) • fun i => φ i x₀
⊢ (support fun i => (ρ i) x₀ • φ i x₀) ⊆ ↑(ρ.finsupport x₀)
state_after: ι : Type u
X : Type v
inst✝⁶ : TopologicalSpace X
E : Type u_1
inst✝⁵ : AddCommMonoid E
inst✝⁴ : SMulWithZero ℝ E
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul ℝ E
s✝ : Set X
f : PartitionOfUnity ι X s✝
s : Set X
ρ : PartitionOfUnity ι X s
x₀✝ x₀ : X
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module ℝ M
φ : ι → X → M
this : (fun i => (ρ i) x₀ • φ i x₀) = (fun i => (ρ i) x₀) • fun i => φ i x₀
⊢ ((support fun i => (ρ i) x₀) ∩ support fun i => φ i x₀) ⊆ support fun i => (ρ i) x₀
tactic: rw [ρ.coe_finsupport x₀, this, support_smul]
state_before: ι : Type u
X : Type v
inst✝⁶ : TopologicalSpace X
E : Type u_1
inst✝⁵ : AddCommMonoid E
inst✝⁴ : SMulWithZero ℝ E
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul ℝ E
s✝ : Set X
f : PartitionOfUnity ι X s✝
s : Set X
ρ : PartitionOfUnity ι X s
x₀✝ x₀ : X
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module ℝ M
φ : ι → X → M
this : (fun i => (ρ i) x₀ • φ i x₀) = (fun i => (ρ i) x₀) • fun i => φ i x₀
⊢ ((support fun i => (ρ i) x₀) ∩ support fun i => φ i x₀) ⊆ support fun i => (ρ i) x₀
state_after: no goals
tactic: exact inter_subset_left