awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (hypothesis)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
⊢ [EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0,
      Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      ∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C, ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C, BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖),
      LT.lt (α := ENNReal) (⨆ (i : ι), (↑‖f i‖₊ : ENNReal)) ⊤].TFAE
have tfae_4_iff_5 : (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
  ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
Tactic: tfae_have 4 ↔ 5 := by
    rw [exists_ge_and_iff_exists]
    exact fun C₁ C₂ hC ↦ forall₂_imp fun i x ↦ le_trans' <| by gcongr
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (goal)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
⊢ (∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
have refine : ∀ (C : ℝ) (hC : C ≥ 0) (i : ι), (∀ (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔ ‖f i‖ ≤ C
Tactic: refine exists_congr (fun C ↦ and_congr_right fun hC ↦ forall_congr' fun i ↦ ?_)
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (goal)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
C : ℝ
hC : C ≥ 0
i : ι
⊢ (∀ (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔ ‖f i‖ ≤ C
have rw : (∀ (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
  ∀ (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
Tactic: rw [ContinuousLinearMap.opNorm_le_iff hC]
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (hypothesis)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
⊢ [EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0,
      Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      ∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C, ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C, BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖),
      LT.lt (α := ENNReal) (⨆ (i : ι), (↑‖f i‖₊ : ENNReal)) ⊤].TFAE
have tfae_5_iff_7 : (∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
Tactic: tfae_have 5 ↔ 7 := by
    refine exists_congr (fun C ↦ and_congr_right fun hC ↦ forall_congr' fun i ↦ ?_)
    rw [ContinuousLinearMap.opNorm_le_iff hC]
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (goal)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
tfae_5_iff_7 :
  (∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
⊢ (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
have simp_rw : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔
  ∃ (x : ℝ), 0 ≤ x ∧ ∀ (y : ℝ), Membership.mem (γ := Set ℝ) (Set.range fun (x : ι) ↦ ‖f x‖) y → y ≤ x
Tactic: simp_rw [bddAbove_iff_exists_ge (0 : ℝ), Set.forall_mem_range]
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (hypothesis)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
tfae_5_iff_7 :
  (∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
⊢ [EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0,
      Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      ∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C, ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C, BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖),
      LT.lt (α := ENNReal) (⨆ (i : ι), (↑‖f i‖₊ : ENNReal)) ⊤].TFAE
have tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
Tactic: tfae_have 7 ↔ 8 := by
    simp_rw [bddAbove_iff_exists_ge (0 : ℝ), Set.forall_mem_range]
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (hypothesis)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
tfae_5_iff_7 :
  (∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
⊢ [EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0,
      Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      ∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C, ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C, BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖),
      LT.lt (α := ENNReal) (⨆ (i : ι), (↑‖f i‖₊ : ENNReal)) ⊤].TFAE
have tfae_6_iff_8 : (∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
Tactic: tfae_have 6 ↔ 8 := by
    simp_rw [bddAbove_def, Set.forall_mem_range]
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (goal)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
tfae_5_iff_7 :
  (∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
tfae_6_iff_8 : (∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
⊢ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖) ↔ LT.lt (α := ENNReal) (⨆ (i : ι), (↑‖f i‖₊ : ENNReal)) ⊤
have rw : BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖) ↔
  BddAbove (Set.range.{0, u_8 + 1} (ι := ι) (NNReal.toReal ∘ fun (i : ι) ↦ ‖f i‖₊))
Tactic: rw [ENNReal.iSup_coe_lt_top, ← NNReal.bddAbove_coe, ← Set.range_comp]
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (hypothesis)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
tfae_5_iff_7 :
  (∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
tfae_6_iff_8 : (∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
⊢ [EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0,
      Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f),
      ∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖,
      ∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C, ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C, BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖),
      LT.lt (α := ENNReal) (⨆ (i : ι), (↑‖f i‖₊ : ENNReal)) ⊤].TFAE
have tfae_8_iff_9 : BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖) ↔ LT.lt (α := ENNReal) (⨆ (i : ι), (↑‖f i‖₊ : ENNReal)) ⊤
Tactic: tfae_have 8 ↔ 9 := by
    rw [ENNReal.iSup_coe_lt_top, ← NNReal.bddAbove_coe, ← Set.range_comp]
    rfl
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (goal)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
tfae_5_iff_7 :
  (∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
tfae_6_iff_8 : (∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
tfae_8_iff_9 :
  BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖) ↔ LT.lt (α := ENNReal) (⨆ (i : ι), (↑‖f i‖₊ : ENNReal)) ⊤
⊢ UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) ↔
    ∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
have refine : (∀ (i : Fin 1),
    ∃ (p : Seminorm 𝕜 E),
      Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] (⇑p : (a : E) → ℝ) ∧
        ∀ (k : ι), Seminorm.comp (normSeminorm 𝕜₂ F) (↑(f k) : E →ₛₗ[σ₁₂] F) ≤ p) ↔
  ∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
Tactic: refine ((norm_withSeminorms 𝕜₂ F).uniformEquicontinuous_iff_exists_continuous_seminorm _).trans
      ?_
-- Lean 4 proof step from Mathlib.Analysis.Normed.Operator.NormedSpace (goal)
Goal:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
ι : Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst_2 : RingHomIsometric σ₁₂
inst_3 : SeminormedAddCommGroup E
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 E
inst_6 : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 :
  ∀ (a : EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0),
    UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_3_to_2 :
  ∀ (a : UniformEquicontinuous.{u_8, u_5, u_4} (α := F) (β := E) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)
tfae_2_to_1 :
  ∀ (a : Equicontinuous.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f)),
    EquicontinuousAt.{u_8, u_4, u_5} (X := E) (α := F) (DFunLike.coe (β := fun (x : E) ↦ F) ∘ f) 0
tfae_4_iff_5 :
  (∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
tfae_5_iff_7 :
  (∃ C ≥ 0, ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖) ↔
    ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
tfae_6_iff_8 : (∃ (C : ℝ), ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖)
tfae_8_iff_9 :
  BddAbove (α := ℝ) (Set.range fun (x : ι) ↦ ‖f x‖) ↔ LT.lt (α := ENNReal) (⨆ (i : ι), (↑‖f i‖₊ : ENNReal)) ⊤
⊢ (∀ (i : Fin 1),
      ∃ (p : Seminorm 𝕜 E),
        Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] (⇑p : (a : E) → ℝ) ∧
          ∀ (k : ι), Seminorm.comp (normSeminorm 𝕜₂ F) (↑(f k) : E →ₛₗ[σ₁₂] F) ≤ p) ↔
    ∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
have rw : (∃ (p : Seminorm 𝕜 E),
    Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] (⇑p : (a : E) → ℝ) ∧
      ∀ (k : ι), Seminorm.comp (normSeminorm 𝕜₂ F) (↑(f k) : E →ₛₗ[σ₁₂] F) ≤ p) ↔
  ∃ (C : ℝ), ∀ (i : ι) (x : E), norm.{u_5} (E := F) ((f i : (a : E) → F) x : F) ≤ C * ‖x‖
Tactic: rw [forall_const]