Theorem: _private.Mathlib.Topology.Algebra.Module.FiniteDimension.0.continuous_equivFun_basis_aux
File: Mathlib/Topology/Algebra/Module/FiniteDimension.lean

Proof steps:
state_before: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : Module 𝕜 E
inst✝¹⁵ : TopologicalSpace E
inst✝¹⁴ : TopologicalAddGroup E
inst✝¹³ : ContinuousSMul 𝕜 E
F : Type w
inst✝¹² : AddCommGroup F
inst✝¹¹ : Module 𝕜 F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : ContinuousSMul 𝕜 F
F' : Type x
inst✝⁷ : AddCommGroup F'
inst✝⁶ : Module 𝕜 F'
inst✝⁵ : TopologicalSpace F'
inst✝⁴ : TopologicalAddGroup F'
inst✝³ : ContinuousSMul 𝕜 F'
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Fintype ι
ξ : Basis ι 𝕜 E
⊢ Continuous ⇑ξ.equivFun
state_after: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : Module 𝕜 E
inst✝¹⁵ : TopologicalSpace E
inst✝¹⁴ : TopologicalAddGroup E
inst✝¹³ : ContinuousSMul 𝕜 E
F : Type w
inst✝¹² : AddCommGroup F
inst✝¹¹ : Module 𝕜 F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : ContinuousSMul 𝕜 F
F' : Type x
inst✝⁷ : AddCommGroup F'
inst✝⁶ : Module 𝕜 F'
inst✝⁵ : TopologicalSpace F'
inst✝⁴ : TopologicalAddGroup F'
inst✝³ : ContinuousSMul 𝕜 F'
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Fintype ι
ξ : Basis ι 𝕜 E
this : UniformSpace E := TopologicalAddGroup.toUniformSpace E
⊢ Continuous ⇑ξ.equivFun
tactic: letI : UniformSpace E := TopologicalAddGroup.toUniformSpace E
state_before: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : Module 𝕜 E
inst✝¹⁵ : TopologicalSpace E
inst✝¹⁴ : TopologicalAddGroup E
inst✝¹³ : ContinuousSMul 𝕜 E
F : Type w
inst✝¹² : AddCommGroup F
inst✝¹¹ : Module 𝕜 F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : ContinuousSMul 𝕜 F
F' : Type x
inst✝⁷ : AddCommGroup F'
inst✝⁶ : Module 𝕜 F'
inst✝⁵ : TopologicalSpace F'
inst✝⁴ : TopologicalAddGroup F'
inst✝³ : ContinuousSMul 𝕜 F'
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Fintype ι
ξ : Basis ι 𝕜 E
this : UniformSpace E := TopologicalAddGroup.toUniformSpace E
⊢ Continuous ⇑ξ.equivFun
state_after: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : Module 𝕜 E
inst✝¹⁵ : TopologicalSpace E
inst✝¹⁴ : TopologicalAddGroup E
inst✝¹³ : ContinuousSMul 𝕜 E
F : Type w
inst✝¹² : AddCommGroup F
inst✝¹¹ : Module 𝕜 F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : ContinuousSMul 𝕜 F
F' : Type x
inst✝⁷ : AddCommGroup F'
inst✝⁶ : Module 𝕜 F'
inst✝⁵ : TopologicalSpace F'
inst✝⁴ : TopologicalAddGroup F'
inst✝³ : ContinuousSMul 𝕜 F'
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Fintype ι
ξ : Basis ι 𝕜 E
this✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this : UniformAddGroup E := comm_topologicalAddGroup_is_uniform
⊢ Continuous ⇑ξ.equivFun
tactic: letI : UniformAddGroup E := comm_topologicalAddGroup_is_uniform
state_before: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : Module 𝕜 E
inst✝¹⁵ : TopologicalSpace E
inst✝¹⁴ : TopologicalAddGroup E
inst✝¹³ : ContinuousSMul 𝕜 E
F : Type w
inst✝¹² : AddCommGroup F
inst✝¹¹ : Module 𝕜 F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : ContinuousSMul 𝕜 F
F' : Type x
inst✝⁷ : AddCommGroup F'
inst✝⁶ : Module 𝕜 F'
inst✝⁵ : TopologicalSpace F'
inst✝⁴ : TopologicalAddGroup F'
inst✝³ : ContinuousSMul 𝕜 F'
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Fintype ι
ξ : Basis ι 𝕜 E
this✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this : UniformAddGroup E := comm_topologicalAddGroup_is_uniform
⊢ Continuous ⇑ξ.equivFun
state_after: case zero
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
F : Type w
inst✝¹⁷ : AddCommGroup F
inst✝¹⁶ : Module 𝕜 F
inst✝¹⁵ : TopologicalSpace F
inst✝¹⁴ : TopologicalAddGroup F
inst✝¹³ : ContinuousSMul 𝕜 F
F' : Type x
inst✝¹² : AddCommGroup F'
inst✝¹¹ : Module 𝕜 F'
inst✝¹⁰ : TopologicalSpace F'
inst✝⁹ : TopologicalAddGroup F'
inst✝⁸ : ContinuousSMul 𝕜 F'
inst✝⁷ : CompleteSpace 𝕜
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Fintype ι
ξ : Basis ι 𝕜 E
this✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this : UniformAddGroup E := comm_topologicalAddGroup_is_uniform
hn : Fintype.card ι = 0
⊢ Continuous ⇑ξ.equivFun

case succ
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
F : Type w
inst✝¹⁷ : AddCommGroup F
inst✝¹⁶ : Module 𝕜 F
inst✝¹⁵ : TopologicalSpace F
inst✝¹⁴ : TopologicalAddGroup F
inst✝¹³ : ContinuousSMul 𝕜 F
F' : Type x
inst✝¹² : AddCommGroup F'
inst✝¹¹ : Module 𝕜 F'
inst✝¹⁰ : TopologicalSpace F'
inst✝⁹ : TopologicalAddGroup F'
inst✝⁸ : ContinuousSMul 𝕜 F'
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : TopologicalAddGroup E] [inst_4 : ContinuousSMul 𝕜 E] [inst_5 : T2Space E] {ι : Type v}
    [inst_6 : Fintype ι] (ξ : Basis ι 𝕜 E),
    let this := TopologicalAddGroup.toUniformSpace E;
    let this := ⋯;
    Fintype.card ι = n → Continuous ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Fintype ι
ξ : Basis ι 𝕜 E
this✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this : UniformAddGroup E := comm_topologicalAddGroup_is_uniform
hn : Fintype.card ι = n + 1
⊢ Continuous ⇑ξ.equivFun
tactic: induction' hn : Fintype.card ι with n IH generalizing ι E