-- Mathlib/Topology/PartitionOfUnity.lean
theorem sum_finsupport' (hx₀ : x₀ ∈ s) {I : Finset ι} (hI : ρ.finsupport x₀ ⊆ I) :
    ∑ i ∈ I, ρ i x₀ = 1

Proof state:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
⊢ ∀ x ∈ I \ ρ.finsupport x₀, (ρ x) x₀ = 0

Tactic:
rintro x hx

Result:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
x : ι
hx : x ∈ I \ ρ.finsupport x₀
⊢ (ρ x) x₀ = 0

Proof state:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
this : ∑ i ∈ I \ ρ.finsupport x₀, (ρ i) x₀ = ∑ i ∈ I \ ρ.finsupport x₀, 0
⊢ ∑ x ∈ I \ ρ.finsupport x₀, (ρ x) x₀ + 1 = 1

Tactic:
rw [this, add_left_eq_self, Finset.sum_const_zero]

Result:
no goals

Proof state:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
⊢ ∑ i ∈ I, (ρ i) x₀ = 1

Tactic:
rw [← Finset.sum_sdiff hI, ρ.sum_finsupport hx₀]

Result:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
⊢ ∑ x ∈ I \ ρ.finsupport x₀, (ρ x) x₀ + 1 = 1

Proof state:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
x : ι
hx : x ∈ I \ ρ.finsupport x₀
⊢ (ρ x) x₀ = 0

Tactic:
simp only [Finset.mem_sdiff, ρ.mem_finsupport, mem_support, Classical.not_not] at hx

Result:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
x : ι
hx : x ∈ I ∧ (ρ x) x₀ = 0
⊢ (ρ x) x₀ = 0

Proof state:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
⊢ ∑ i ∈ I, (ρ i) x₀ = 1

Tactic:
classical
rw [← Finset.sum_sdiff hI, ρ.sum_finsupport hx₀]
suffices ∑ i ∈ I \ ρ.finsupport x₀, (ρ i) x₀ = ∑ i ∈ I \ ρ.finsupport x₀, 0 by
  rw [this, add_left_eq_self, Finset.sum_const_zero]
apply Finset.sum_congr rfl
rintro x hx
simp only [Finset.mem_sdiff, ρ.mem_finsupport, mem_support, Classical.not_not] at hx
exact hx.2

Result:
no goals

Proof state:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
⊢ ∑ i ∈ I \ ρ.finsupport x₀, (ρ i) x₀ = ∑ i ∈ I \ ρ.finsupport x₀, 0

Tactic:
apply Finset.sum_congr rfl

Result:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
⊢ ∀ x ∈ I \ ρ.finsupport x₀, (ρ x) x₀ = 0

Proof state:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
⊢ ∑ x ∈ I \ ρ.finsupport x₀, (ρ x) x₀ + 1 = 1

Tactic:
suffices ∑ i ∈ I \ ρ.finsupport x₀, (ρ i) x₀ = ∑ i ∈ I \ ρ.finsupport x₀, 0 by
  rw [this, add_left_eq_self, Finset.sum_const_zero]

Result:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
⊢ ∑ i ∈ I \ ρ.finsupport x₀, (ρ i) x₀ = ∑ i ∈ I \ ρ.finsupport x₀, 0

Proof state:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
hx₀ : x₀ ∈ s
I : Finset ι
hI : ρ.finsupport x₀ ⊆ I
x : ι
hx : x ∈ I ∧ (ρ x) x₀ = 0
⊢ (ρ x) x₀ = 0

Tactic:
exact hx.2

Result:
no goals