/-- Equivalent characterizations for equicontinuity of a family of continuous linear maps
between normed spaces. See also `WithSeminorms.equicontinuous_TFAE` for similar characterizations
between spaces satisfying `WithSeminorms`. -/
protected theorem NormedSpace.equicontinuous_TFAE : List.TFAE
    [ EquicontinuousAt ((↑) ∘ f) 0,
      Equicontinuous ((↑) ∘ f),
      UniformEquicontinuous ((↑) ∘ f),
      ∃ C, ∀ i x, ‖f i x‖ ≤ C * ‖x‖,
      ∃ C ≥ 0, ∀ i x, ‖f i x‖ ≤ C * ‖x‖,
      ∃ C, ∀ i, ‖f i‖ ≤ C,
      ∃ C ≥ 0, ∀ i, ‖f i‖ ≤ C,
      BddAbove (Set.range (‖f ·‖)),
      (⨆ i, (‖f i‖₊ : ENNReal)) < ⊤ ] := by
  -- `1 ↔ 2 ↔ 3` follows from `uniformEquicontinuous_of_equicontinuousAt_zero`
  tfae_have 1 → 3
  · exact uniformEquicontinuous_of_equicontinuousAt_zero f
  tfae_have 3 → 2
  · exact UniformEquicontinuous.equicontinuous
  tfae_have 2 → 1
  · exact fun H ↦ H 0
  -- `4 ↔ 5 ↔ 6 ↔ 7 ↔ 8 ↔ 9` is morally trivial, we just have to use a lot of rewriting
  -- and `congr` lemmas
  tfae_have 4 ↔ 5
  · rw [exists_ge_and_iff_exists]
    exact fun C₁ C₂ hC ↦ forall₂_imp fun i x ↦ le_trans' <| by gcongr
  tfae_have 5 ↔ 7
  · refine exists_congr (fun C ↦ and_congr_right fun hC ↦ forall_congr' fun i ↦ ?_)
    rw [ContinuousLinearMap.op_norm_le_iff hC]
  tfae_have 7 ↔ 8
  · simp_rw [bddAbove_iff_exists_ge (0 : ℝ), Set.forall_range_iff]
  tfae_have 6 ↔ 8
  · simp_rw [bddAbove_def, Set.forall_range_iff]
  tfae_have 8 ↔ 9
  · rw [ENNReal.iSup_coe_lt_top, ← NNReal.bddAbove_coe, ← Set.range_comp]
    rfl
  -- `3 ↔ 4` is the interesting part of the result. It is essentially a combination of
  -- `WithSeminorms.uniformEquicontinuous_iff_exists_continuous_seminorm` which turns
  -- equicontinuity into existence of some continuous seminorm and
  -- `Seminorm.bound_of_continuous_normedSpace` which characterize such seminorms.
  tfae_have 3 ↔ 4
  · refine ((norm_withSeminorms 𝕜₂ F).uniformEquicontinuous_iff_exists_continuous_seminorm _).trans
      ?_
    rw [forall_const]
    constructor
    · intro ⟨p, hp, hpf⟩
      rcases p.bound_of_continuous_normedSpace hp with ⟨C, -, hC⟩
      exact ⟨C, fun i x ↦ (hpf i x).trans (hC x)⟩
    · intro ⟨C, hC⟩
      refine ⟨C.toNNReal • normSeminorm 𝕜 E,
        ((norm_withSeminorms 𝕜 E).continuous_seminorm 0).const_smul C.toNNReal, fun i x ↦ ?_⟩
      

Proof state:
case tfae_3_iff_4.mpr
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
ι : Type u_11
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 : EquicontinuousAt (FunLike.coe ∘ f) 0 → UniformEquicontinuous (FunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (FunLike.coe ∘ f) → Equicontinuous (FunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (FunLike.coe ∘ f) → EquicontinuousAt (FunLike.coe ∘ f) 0
tfae_4_iff_5 : (∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖
tfae_5_iff_7 : (∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x => ‖f x‖)
tfae_6_iff_8 : (∃ C, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x => ‖f x‖)
tfae_8_iff_9 : BddAbove (Set.range fun x => ‖f x‖) ↔ ⨆ i, ↑‖f i‖₊ < ⊤
C : ℝ
hC : ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖
i : ι
x : E
⊢ (fun f => ⇑f) (Seminorm.comp (normSeminorm 𝕜₂ F) ↑(f i)) x ≤ (fun f => ⇑f) (Real.toNNReal C • normSeminorm 𝕜 E) x

Next tactic:
refine (hC i x).trans (mul_le_mul_of_nonneg_right (C.le_coe_toNNReal) (norm_nonneg x))