Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝³ : RingHomIsometric σ₁₂
κ : Type u_11
q : SeminormFamily 𝕜₂ F ι'
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousSMul 𝕜 E
f : κ → E →ₛₗ[σ₁₂] F
i : ι'
x✝ : SeminormedAddCommGroup F := (q i).toSeminormedAddCommGroup
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
tfae_3_to_5 :
  UniformEquicontinuous (DFunLike.coe ∘ f) →
    BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧ Continuous[inst✝².toTopologicalSpace, _] (⨆ k, ⇑((q i).comp (f k)))
tfae_5_to_4 :
  BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧ Continuous[inst✝².toTopologicalSpace, _] (⨆ k, ⇑((q i).comp (f k))) →
    ∃ p, Continuous[inst✝².toTopologicalSpace, _] ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p
⊢ [EquicontinuousAt (DFunLike.coe ∘ f) 0, Equicontinuous (DFunLike.coe ∘ f), UniformEquicontinuous (DFunLike.coe ∘ f),
      ∃ p, Continuous[inst✝².toTopologicalSpace, _] ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p,
      BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧
        Continuous[inst✝².toTopologicalSpace, _] (⨆ k, ⇑((q i).comp (f k)))].TFAE

Tactic:
tfae_have 4 → 1 -- This would work over any `NormedField`
  | ⟨p, hp, hfp⟩ =>
    Metric.equicontinuousAt_of_continuity_modulus p (map_zero p ▸ hp.tendsto 0) _ <|
      Eventually.of_forall fun x k ↦ by simpa using! hfp k x

Goal state after:
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝³ : RingHomIsometric σ₁₂
κ : Type u_11
q : SeminormFamily 𝕜₂ F ι'
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousSMul 𝕜 E
f : κ → E →ₛₗ[σ₁₂] F
i : ι'
x✝ : SeminormedAddCommGroup F := (q i).toSeminormedAddCommGroup
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
tfae_3_to_5 :
  UniformEquicontinuous (DFunLike.coe ∘ f) →
    BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧ Continuous[inst✝².toTopologicalSpace, _] (⨆ k, ⇑((q i).comp (f k)))
tfae_5_to_4 :
  BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧ Continuous[inst✝².toTopologicalSpace, _] (⨆ k, ⇑((q i).comp (f k))) →
    ∃ p, Continuous[inst✝².toTopologicalSpace, _] ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p
tfae_4_to_1 :
  (∃ p, Continuous[inst✝².toTopologicalSpace, _] ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p) →
    EquicontinuousAt (DFunLike.coe ∘ f) 0
⊢ [EquicontinuousAt (DFunLike.coe ∘ f) 0, Equicontinuous (DFunLike.coe ∘ f), UniformEquicontinuous (DFunLike.coe ∘ f),
      ∃ p, Continuous[inst✝².toTopologicalSpace, _] ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p,
      BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧
        Continuous[inst✝².toTopologicalSpace, _] (⨆ k, ⇑((q i).comp (f k)))].TFAE