Theorem: ContinuousSMul.induced
File: Mathlib/Topology/Algebra/MulAction.lean

Proof steps:
state_before: M : Type u_1
X : Type u_2
Y : Type u_3
α✝ : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : TopologicalSpace X
inst✝¹¹ : TopologicalSpace Y
inst✝¹⁰ : SMul M X
inst✝⁹ : ContinuousSMul M X
R : Type u_5
α : Type u_6
β : Type u_7
F : Type u_8
inst✝⁸ : FunLike F α β
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid α
inst✝⁵ : AddCommMonoid β
inst✝⁴ : Module R α
inst✝³ : Module R β
inst✝² : TopologicalSpace R
inst✝¹ : LinearMapClass F R α β
tβ : TopologicalSpace β
inst✝ : ContinuousSMul R β
f : F
⊢ ContinuousSMul R α
state_after: M : Type u_1
X : Type u_2
Y : Type u_3
α✝ : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : TopologicalSpace X
inst✝¹¹ : TopologicalSpace Y
inst✝¹⁰ : SMul M X
inst✝⁹ : ContinuousSMul M X
R : Type u_5
α : Type u_6
β : Type u_7
F : Type u_8
inst✝⁸ : FunLike F α β
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid α
inst✝⁵ : AddCommMonoid β
inst✝⁴ : Module R α
inst✝³ : Module R β
inst✝² : TopologicalSpace R
inst✝¹ : LinearMapClass F R α β
tβ : TopologicalSpace β
inst✝ : ContinuousSMul R β
f : F
tα : TopologicalSpace α := TopologicalSpace.induced (⇑f) tβ
⊢ ContinuousSMul R α
tactic: let tα := tβ.induced f
state_before: M : Type u_1
X : Type u_2
Y : Type u_3
α✝ : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : TopologicalSpace X
inst✝¹¹ : TopologicalSpace Y
inst✝¹⁰ : SMul M X
inst✝⁹ : ContinuousSMul M X
R : Type u_5
α : Type u_6
β : Type u_7
F : Type u_8
inst✝⁸ : FunLike F α β
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid α
inst✝⁵ : AddCommMonoid β
inst✝⁴ : Module R α
inst✝³ : Module R β
inst✝² : TopologicalSpace R
inst✝¹ : LinearMapClass F R α β
tβ : TopologicalSpace β
inst✝ : ContinuousSMul R β
f : F
tα : TopologicalSpace α := TopologicalSpace.induced (⇑f) tβ
⊢ ContinuousSMul R α
state_after: M : Type u_1
X : Type u_2
Y : Type u_3
α✝ : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : TopologicalSpace X
inst✝¹¹ : TopologicalSpace Y
inst✝¹⁰ : SMul M X
inst✝⁹ : ContinuousSMul M X
R : Type u_5
α : Type u_6
β : Type u_7
F : Type u_8
inst✝⁸ : FunLike F α β
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid α
inst✝⁵ : AddCommMonoid β
inst✝⁴ : Module R α
inst✝³ : Module R β
inst✝² : TopologicalSpace R
inst✝¹ : LinearMapClass F R α β
tβ : TopologicalSpace β
inst✝ : ContinuousSMul R β
f : F
tα : TopologicalSpace α := TopologicalSpace.induced (⇑f) tβ
⊢ Continuous (⇑f ∘ fun p => p.1 • p.2)
tactic: refine ⟨continuous_induced_rng.2 ?_⟩
state_before: M : Type u_1
X : Type u_2
Y : Type u_3
α✝ : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : TopologicalSpace X
inst✝¹¹ : TopologicalSpace Y
inst✝¹⁰ : SMul M X
inst✝⁹ : ContinuousSMul M X
R : Type u_5
α : Type u_6
β : Type u_7
F : Type u_8
inst✝⁸ : FunLike F α β
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid α
inst✝⁵ : AddCommMonoid β
inst✝⁴ : Module R α
inst✝³ : Module R β
inst✝² : TopologicalSpace R
inst✝¹ : LinearMapClass F R α β
tβ : TopologicalSpace β
inst✝ : ContinuousSMul R β
f : F
tα : TopologicalSpace α := TopologicalSpace.induced (⇑f) tβ
⊢ Continuous (⇑f ∘ fun p => p.1 • p.2)
state_after: M : Type u_1
X : Type u_2
Y : Type u_3
α✝ : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : TopologicalSpace X
inst✝¹¹ : TopologicalSpace Y
inst✝¹⁰ : SMul M X
inst✝⁹ : ContinuousSMul M X
R : Type u_5
α : Type u_6
β : Type u_7
F : Type u_8
inst✝⁸ : FunLike F α β
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid α
inst✝⁵ : AddCommMonoid β
inst✝⁴ : Module R α
inst✝³ : Module R β
inst✝² : TopologicalSpace R
inst✝¹ : LinearMapClass F R α β
tβ : TopologicalSpace β
inst✝ : ContinuousSMul R β
f : F
tα : TopologicalSpace α := TopologicalSpace.induced (⇑f) tβ
⊢ Continuous fun x => x.1 • f x.2
tactic: simp only [Function.comp, map_smul]
state_before: M : Type u_1
X : Type u_2
Y : Type u_3
α✝ : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : TopologicalSpace X
inst✝¹¹ : TopologicalSpace Y
inst✝¹⁰ : SMul M X
inst✝⁹ : ContinuousSMul M X
R : Type u_5
α : Type u_6
β : Type u_7
F : Type u_8
inst✝⁸ : FunLike F α β
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid α
inst✝⁵ : AddCommMonoid β
inst✝⁴ : Module R α
inst✝³ : Module R β
inst✝² : TopologicalSpace R
inst✝¹ : LinearMapClass F R α β
tβ : TopologicalSpace β
inst✝ : ContinuousSMul R β
f : F
tα : TopologicalSpace α := TopologicalSpace.induced (⇑f) tβ
⊢ Continuous fun x => x.1 • f x.2
state_after: no goals
tactic: fun_prop