-- Mathlib/Topology/Algebra/MulAction.lean
theorem ContinuousSMul.induced {R : Type*} {α : Type*} {β : Type*} {F : Type*} [FunLike F α β]
    [Semiring R] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β]
    [TopologicalSpace R] [LinearMapClass F R α β] [tβ : TopologicalSpace β] [ContinuousSMul R β]
    (f : F) : @ContinuousSMul R α _ _ (tβ.induced f)

Proof state:
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:
refine ⟨continuous_induced_rng.2 ?_⟩

Result:
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)

Proof state:
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

Tactic:
let tα := tβ.induced f

Result:
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

Proof state:
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:
fun_prop

Result:
no goals

Proof state:
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:
simp only [Function.comp_def, map_smul]

Result:
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