Theorem: mul_mem_connectedComponent_one
File: Mathlib/Topology/Algebra/Group/Basic.lean

tactic: rw [connectedComponent_eq hg]
state_before: α : Type u
β : Type v
G✝ : Type w
H : Type x
inst✝⁶ : TopologicalSpace G✝
inst✝⁵ : Group G✝
inst✝⁴ : TopologicalGroup G✝
inst✝³ : TopologicalSpace α
f : α → G✝
s : Set α
x : α
G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
g h : G
hg : g ∈ connectedComponent 1
hh : h ∈ connectedComponent 1
⊢ g * h ∈ connectedComponent 1

tactic: have hmul : g ∈ connectedComponent (g * h) := by
  apply Continuous.image_connectedComponent_subset (continuous_mul_left g)
  rw [← connectedComponent_eq hh]
  exact ⟨(1 : G), mem_connectedComponent, by simp only [mul_one]⟩
state_before: α : Type u
β : Type v
G✝ : Type w
H : Type x
inst✝⁶ : TopologicalSpace G✝
inst✝⁵ : Group G✝
inst✝⁴ : TopologicalGroup G✝
inst✝³ : TopologicalSpace α
f : α → G✝
s : Set α
x : α
G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
g h : G
hg : g ∈ connectedComponent 1
hh : h ∈ connectedComponent 1
⊢ g * h ∈ connectedComponent g

tactic: simpa [← connectedComponent_eq hmul] using mem_connectedComponent
state_before: α : Type u
β : Type v
G✝ : Type w
H : Type x
inst✝⁶ : TopologicalSpace G✝
inst✝⁵ : Group G✝
inst✝⁴ : TopologicalGroup G✝
inst✝³ : TopologicalSpace α
f : α → G✝
s : Set α
x : α
G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
g h : G
hg : g ∈ connectedComponent 1
hh : h ∈ connectedComponent 1
hmul : g ∈ connectedComponent (g * h)
⊢ g * h ∈ connectedComponent g

tactic: apply Continuous.image_connectedComponent_subset (continuous_mul_left g)
state_before: α : Type u
β : Type v
G✝ : Type w
H : Type x
inst✝⁶ : TopologicalSpace G✝
inst✝⁵ : Group G✝
inst✝⁴ : TopologicalGroup G✝
inst✝³ : TopologicalSpace α
f : α → G✝
s : Set α
x : α
G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
g h : G
hg : g ∈ connectedComponent 1
hh : h ∈ connectedComponent 1
⊢ g ∈ connectedComponent (g * h)

tactic: rw [← connectedComponent_eq hh]
state_before: case a
α : Type u
β : Type v
G✝ : Type w
H : Type x
inst✝⁶ : TopologicalSpace G✝
inst✝⁵ : Group G✝
inst✝⁴ : TopologicalGroup G✝
inst✝³ : TopologicalSpace α
f : α → G✝
s : Set α
x : α
G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
g h : G
hg : g ∈ connectedComponent 1
hh : h ∈ connectedComponent 1
⊢ g ∈ (fun b => g * b) '' connectedComponent h

tactic: exact ⟨(1 : G), mem_connectedComponent, by simp only [mul_one]⟩
state_before: case a
α : Type u
β : Type v
G✝ : Type w
H : Type x
inst✝⁶ : TopologicalSpace G✝
inst✝⁵ : Group G✝
inst✝⁴ : TopologicalGroup G✝
inst✝³ : TopologicalSpace α
f : α → G✝
s : Set α
x : α
G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
g h : G
hg : g ∈ connectedComponent 1
hh : h ∈ connectedComponent 1
⊢ g ∈ (fun b => g * b) '' connectedComponent 1

tactic: simp only [mul_one]
state_before: α : Type u
β : Type v
G✝ : Type w
H : Type x
inst✝⁶ : TopologicalSpace G✝
inst✝⁵ : Group G✝
inst✝⁴ : TopologicalGroup G✝
inst✝³ : TopologicalSpace α
f : α → G✝
s : Set α
x : α
G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : MulOneClass G
inst✝ : ContinuousMul G
g h : G
hg : g ∈ connectedComponent 1
hh : h ∈ connectedComponent 1
⊢ (fun b => g * b) 1 = g