Theorem: ContinuousMul.of_nhds_one
File: Mathlib/Topology/Algebra/Monoid.lean

tactic: rw [continuous_iff_continuousAt]
state_before: ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
⊢ Continuous fun p => p.fst * p.snd

tactic: rintro ⟨x₀, y₀⟩
state_before: ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
⊢ ∀ (x : M × M), ContinuousAt (fun p => p.fst * p.snd) x

tactic: have key₂ : ((fun x => x₀ * x) ∘ fun x => y₀ * x) = fun x => x₀ * y₀ * x := by
  ext x
  simp [mul_assoc]
state_before: case mk
ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p => x₀ * p.fst * (p.snd * y₀)) = ((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry fun x x_1 => x * x_1
⊢ ContinuousAt (fun p => p.fst * p.snd) (x₀, y₀)

tactic: ext p
state_before: ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
x₀ y₀ : M
⊢ (fun p => x₀ * p.fst * (p.snd * y₀)) = ((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry fun x x_1 => x * x_1

tactic: simp [uncurry, mul_assoc]
state_before: case h
ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
x₀ y₀ : M
p : M × M
⊢ x₀ * p.fst * (p.snd * y₀) = (((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry fun x x_1 => x * x_1) p

tactic: ext x
state_before: ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p => x₀ * p.fst * (p.snd * y₀)) = ((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry fun x x_1 => x * x_1
⊢ ((fun x => x₀ * x) ∘ fun x => y₀ * x) = fun x => x₀ * y₀ * x

tactic: simp [mul_assoc]
state_before: case h
ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p => x₀ * p.fst * (p.snd * y₀)) = ((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry fun x x_1 => x * x_1
x : M
⊢ ((fun x => x₀ * x) ∘ fun x => y₀ * x) x = x₀ * y₀ * x

tactic: rw [nhds_prod_eq]
state_before: ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p => x₀ * p.fst * (p.snd * y₀)) = ((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry fun x x_1 => x * x_1
key₂ : ((fun x => x₀ * x) ∘ fun x => y₀ * x) = fun x => x₀ * y₀ * x
⊢ map (uncurry fun x x_1 => x * x_1) (𝓝 (x₀, y₀)) = map (uncurry fun x x_1 => x * x_1) (𝓝 x₀ ×ˢ 𝓝 y₀)

tactic: simp_rw [uncurry, hleft x₀, hright y₀, prod_map_map_eq, Filter.map_map, Function.comp]
state_before: ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p => x₀ * p.fst * (p.snd * y₀)) = ((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry fun x x_1 => x * x_1
key₂ : ((fun x => x₀ * x) ∘ fun x => y₀ * x) = fun x => x₀ * y₀ * x
⊢ map (uncurry fun x x_1 => x * x_1) (𝓝 x₀ ×ˢ 𝓝 y₀) = map (fun p => x₀ * p.fst * (p.snd * y₀)) (𝓝 1 ×ˢ 𝓝 1)

tactic: rw [key, ← Filter.map_map]
state_before: ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p => x₀ * p.fst * (p.snd * y₀)) = ((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry fun x x_1 => x * x_1
key₂ : ((fun x => x₀ * x) ∘ fun x => y₀ * x) = fun x => x₀ * y₀ * x
⊢ map (fun p => x₀ * p.fst * (p.snd * y₀)) (𝓝 1 ×ˢ 𝓝 1) =
    map ((fun x => x₀ * x) ∘ fun x => x * y₀) (map (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1))

tactic: rw [← Filter.map_map, ← hright, hleft y₀, Filter.map_map, key₂, ← hleft]
state_before: ι : Type ?u.47046
α : Type ?u.47049
X : Type ?u.47052
M✝ : Type ?u.47055
N : Type ?u.47058
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hright : ∀ (x₀ : M), 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)
x₀ y₀ : M
key : (fun p => x₀ * p.fst * (p.snd * y₀)) = ((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry fun x x_1 => x * x_1
key₂ : ((fun x => x₀ * x) ∘ fun x => y₀ * x) = fun x => x₀ * y₀ * x
⊢ map ((fun x => x₀ * x) ∘ fun x => x * y₀) (𝓝 1) = 𝓝 (x₀ * y₀)