Theorem: Flow.ext
File: Mathlib/Dynamics/Flow.lean

Proof steps:
state_before: τ : Type u_1
inst✝³ : AddMonoid τ
inst✝² : TopologicalSpace τ
inst✝¹ : ContinuousAdd τ
α : Type u_2
inst✝ : TopologicalSpace α
ϕ : Flow τ α
f₁ : τ → α → α
cont'✝¹ : Continuous (uncurry f₁)
map_add'✝¹ : ∀ (t₁ t₂ : τ) (x : α), f₁ (t₁ + t₂) x = f₁ t₁ (f₁ t₂ x)
map_zero'✝¹ : ∀ (x : α), f₁ 0 x = x
f₂ : τ → α → α
cont'✝ : Continuous (uncurry f₂)
map_add'✝ : ∀ (t₁ t₂ : τ) (x : α), f₂ (t₁ + t₂) x = f₂ t₁ (f₂ t₂ x)
map_zero'✝ : ∀ (x : α), f₂ 0 x = x
h :
  ∀ (t : τ) (x : α),
    { toFun := f₁, cont' := cont'✝¹, map_add' := map_add'✝¹, map_zero' := map_zero'✝¹ }.toFun t x =
      { toFun := f₂, cont' := cont'✝, map_add' := map_add'✝, map_zero' := map_zero'✝ }.toFun t x
⊢ { toFun := f₁, cont' := cont'✝¹, map_add' := map_add'✝¹, map_zero' := map_zero'✝¹ } =
    { toFun := f₂, cont' := cont'✝, map_add' := map_add'✝, map_zero' := map_zero'✝ }
state_after: case e_toFun
τ : Type u_1
inst✝³ : AddMonoid τ
inst✝² : TopologicalSpace τ
inst✝¹ : ContinuousAdd τ
α : Type u_2
inst✝ : TopologicalSpace α
ϕ : Flow τ α
f₁ : τ → α → α
cont'✝¹ : Continuous (uncurry f₁)
map_add'✝¹ : ∀ (t₁ t₂ : τ) (x : α), f₁ (t₁ + t₂) x = f₁ t₁ (f₁ t₂ x)
map_zero'✝¹ : ∀ (x : α), f₁ 0 x = x
f₂ : τ → α → α
cont'✝ : Continuous (uncurry f₂)
map_add'✝ : ∀ (t₁ t₂ : τ) (x : α), f₂ (t₁ + t₂) x = f₂ t₁ (f₂ t₂ x)
map_zero'✝ : ∀ (x : α), f₂ 0 x = x
h :
  ∀ (t : τ) (x : α),
    { toFun := f₁, cont' := cont'✝¹, map_add' := map_add'✝¹, map_zero' := map_zero'✝¹ }.toFun t x =
      { toFun := f₂, cont' := cont'✝, map_add' := map_add'✝, map_zero' := map_zero'✝ }.toFun t x
⊢ f₁ = f₂
tactic: congr
state_before: case e_toFun
τ : Type u_1
inst✝³ : AddMonoid τ
inst✝² : TopologicalSpace τ
inst✝¹ : ContinuousAdd τ
α : Type u_2
inst✝ : TopologicalSpace α
ϕ : Flow τ α
f₁ : τ → α → α
cont'✝¹ : Continuous (uncurry f₁)
map_add'✝¹ : ∀ (t₁ t₂ : τ) (x : α), f₁ (t₁ + t₂) x = f₁ t₁ (f₁ t₂ x)
map_zero'✝¹ : ∀ (x : α), f₁ 0 x = x
f₂ : τ → α → α
cont'✝ : Continuous (uncurry f₂)
map_add'✝ : ∀ (t₁ t₂ : τ) (x : α), f₂ (t₁ + t₂) x = f₂ t₁ (f₂ t₂ x)
map_zero'✝ : ∀ (x : α), f₂ 0 x = x
h :
  ∀ (t : τ) (x : α),
    { toFun := f₁, cont' := cont'✝¹, map_add' := map_add'✝¹, map_zero' := map_zero'✝¹ }.toFun t x =
      { toFun := f₂, cont' := cont'✝, map_add' := map_add'✝, map_zero' := map_zero'✝ }.toFun t x
⊢ f₁ = f₂
state_after: case e_toFun.h.h
τ : Type u_1
inst✝³ : AddMonoid τ
inst✝² : TopologicalSpace τ
inst✝¹ : ContinuousAdd τ
α : Type u_2
inst✝ : TopologicalSpace α
ϕ : Flow τ α
f₁ : τ → α → α
cont'✝¹ : Continuous (uncurry f₁)
map_add'✝¹ : ∀ (t₁ t₂ : τ) (x : α), f₁ (t₁ + t₂) x = f₁ t₁ (f₁ t₂ x)
map_zero'✝¹ : ∀ (x : α), f₁ 0 x = x
f₂ : τ → α → α
cont'✝ : Continuous (uncurry f₂)
map_add'✝ : ∀ (t₁ t₂ : τ) (x : α), f₂ (t₁ + t₂) x = f₂ t₁ (f₂ t₂ x)
map_zero'✝ : ∀ (x : α), f₂ 0 x = x
h :
  ∀ (t : τ) (x : α),
    { toFun := f₁, cont' := cont'✝¹, map_add' := map_add'✝¹, map_zero' := map_zero'✝¹ }.toFun t x =
      { toFun := f₂, cont' := cont'✝, map_add' := map_add'✝, map_zero' := map_zero'✝ }.toFun t x
x✝¹ : τ
x✝ : α
⊢ f₁ x✝¹ x✝ = f₂ x✝¹ x✝
tactic: funext
state_before: case e_toFun.h.h
τ : Type u_1
inst✝³ : AddMonoid τ
inst✝² : TopologicalSpace τ
inst✝¹ : ContinuousAdd τ
α : Type u_2
inst✝ : TopologicalSpace α
ϕ : Flow τ α
f₁ : τ → α → α
cont'✝¹ : Continuous (uncurry f₁)
map_add'✝¹ : ∀ (t₁ t₂ : τ) (x : α), f₁ (t₁ + t₂) x = f₁ t₁ (f₁ t₂ x)
map_zero'✝¹ : ∀ (x : α), f₁ 0 x = x
f₂ : τ → α → α
cont'✝ : Continuous (uncurry f₂)
map_add'✝ : ∀ (t₁ t₂ : τ) (x : α), f₂ (t₁ + t₂) x = f₂ t₁ (f₂ t₂ x)
map_zero'✝ : ∀ (x : α), f₂ 0 x = x
h :
  ∀ (t : τ) (x : α),
    { toFun := f₁, cont' := cont'✝¹, map_add' := map_add'✝¹, map_zero' := map_zero'✝¹ }.toFun t x =
      { toFun := f₂, cont' := cont'✝, map_add' := map_add'✝, map_zero' := map_zero'✝ }.toFun t x
x✝¹ : τ
x✝ : α
⊢ f₁ x✝¹ x✝ = f₂ x✝¹ x✝
state_after: no goals
tactic: exact h _ _