Theorem: LocallyFinite.exists_finset_mulSupport
File: Mathlib/Topology/Algebra/Monoid.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
⊢ ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ ↑I
state_after: case intro.intro
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
U : Set X
hxU : U ∈ 𝓝 x₀
hUf : {i | ((fun i => mulSupport (f i)) i ∩ U).Nonempty}.Finite
⊢ ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ ↑I
tactic: rcases hf x₀ with ⟨U, hxU, hUf⟩
state_before: case intro.intro
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
U : Set X
hxU : U ∈ 𝓝 x₀
hUf : {i | ((fun i => mulSupport (f i)) i ∩ U).Nonempty}.Finite
⊢ ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ ↑I
state_after: case intro.intro
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
U : Set X
hxU : U ∈ 𝓝 x₀
hUf : {i | ((fun i => mulSupport (f i)) i ∩ U).Nonempty}.Finite
y : X
hy : y ∈ U
i : ι
hi : i ∈ mulSupport fun i => f i y
⊢ i ∈ ↑hUf.toFinset
tactic: refine ⟨hUf.toFinset, mem_of_superset hxU fun y hy i hi => ?_⟩
state_before: case intro.intro
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
U : Set X
hxU : U ∈ 𝓝 x₀
hUf : {i | ((fun i => mulSupport (f i)) i ∩ U).Nonempty}.Finite
y : X
hy : y ∈ U
i : ι
hi : i ∈ mulSupport fun i => f i y
⊢ i ∈ ↑hUf.toFinset
state_after: case intro.intro
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
U : Set X
hxU : U ∈ 𝓝 x₀
hUf : {i | ((fun i => mulSupport (f i)) i ∩ U).Nonempty}.Finite
y : X
hy : y ∈ U
i : ι
hi : i ∈ mulSupport fun i => f i y
⊢ i ∈ {i | ((fun i => mulSupport (f i)) i ∩ U).Nonempty}
tactic: rw [hUf.coe_toFinset]
state_before: case intro.intro
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
U : Set X
hxU : U ∈ 𝓝 x₀
hUf : {i | ((fun i => mulSupport (f i)) i ∩ U).Nonempty}.Finite
y : X
hy : y ∈ U
i : ι
hi : i ∈ mulSupport fun i => f i y
⊢ i ∈ {i | ((fun i => mulSupport (f i)) i ∩ U).Nonempty}
state_after: no goals
tactic: exact ⟨y, hi, hy⟩