awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst : NontriviallyNormedField 𝕜
inst_1 : (i : ι) → SeminormedAddCommGroup (E i)
inst_2 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_3 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_4 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_5 : SeminormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
f : (i : ι) → LinearIsometryEquiv.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
this :
  g =
    compContinuousLinearMap (M₁ := E₁) (M₁' := E)
      (g.compContinuousLinearMap (M₁ := E) (M₁' := E₁) fun (i : ι) ↦ (↑(↑(f i) : E i ≃L[𝕜] E₁ i) : E i →L[𝕜] E₁ i))
      fun (i : ι) ↦ (↑(↑(f i).symm : E₁ i ≃L[𝕜] E i) : E₁ i →L[𝕜] E i)
rw :
  norm (E := ContinuousMultilinearMap 𝕜 E₁ G)
    (compContinuousLinearMap (M₁ := E₁) (M₁' := E)
      (g.compContinuousLinearMap (M₁ := E) (M₁' := E₁) fun (i : ι) ↦ (↑(↑(f i) : E i ≃L[𝕜] E₁ i) : E i →L[𝕜] E₁ i))
      fun (i : ι) ↦ (↑(↑(f i).symm : E₁ i ≃L[𝕜] E i) : E₁ i →L[𝕜] E i))
| ‖g‖

Tactic:
rw [this]
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst : NontriviallyNormedField 𝕜
inst_1 : (i : ι) → SeminormedAddCommGroup (E i)
inst_2 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_3 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_4 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_5 : SeminormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
f : (i : ι) → LinearIsometryEquiv.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
this :
  g =
    compContinuousLinearMap (M₁ := E₁) (M₁' := E)
      (g.compContinuousLinearMap (M₁ := E) (M₁' := E₁) fun (i : ι) ↦ (↑(↑(f i) : E i ≃L[𝕜] E₁ i) : E i →L[𝕜] E₁ i))
      fun (i : ι) ↦ (↑(↑(f i).symm : E₁ i ≃L[𝕜] E i) : E₁ i →L[𝕜] E i)
conv_lhs :
  norm (E := ContinuousMultilinearMap 𝕜 E₁ G)
      (compContinuousLinearMap (M₁ := E₁) (M₁' := E)
        (g.compContinuousLinearMap (M₁ := E) (M₁' := E₁) fun (i : ι) ↦ (↑(↑(f i) : E i ≃L[𝕜] E₁ i) : E i →L[𝕜] E₁ i))
        fun (i : ι) ↦ (↑(↑(f i).symm : E₁ i ≃L[𝕜] E i) : E₁ i →L[𝕜] E i)) ≤
    norm (E := ContinuousMultilinearMap 𝕜 E G)
      (g.compContinuousLinearMap (M₁ := E) (M₁' := E₁) fun (i : ι) ↦ (f i).toLinearIsometry.toContinuousLinearMap)
⊢ ‖g‖ ≤
    norm (E := ContinuousMultilinearMap 𝕜 E G)
      (g.compContinuousLinearMap (M₁ := E) (M₁' := E₁) fun (i : ι) ↦ (f i).toLinearIsometry.toContinuousLinearMap)

Tactic:
conv_lhs => rw [this]
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst : NontriviallyNormedField 𝕜
inst_1 : (i : ι) → SeminormedAddCommGroup (E i)
inst_2 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_3 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_4 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_5 : SeminormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
f : (i : ι) → LinearIsometryEquiv.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
this :
  g =
    compContinuousLinearMap (M₁ := E₁) (M₁' := E)
      (g.compContinuousLinearMap (M₁ := E) (M₁' := E₁) fun (i : ι) ↦ (↑(↑(f i) : E i ≃L[𝕜] E₁ i) : E i →L[𝕜] E₁ i))
      fun (i : ι) ↦ (↑(↑(f i).symm : E₁ i ≃L[𝕜] E i) : E₁ i →L[𝕜] E i)
⊢ norm (E := ContinuousMultilinearMap 𝕜 E₁ G)
      (compContinuousLinearMap (M₁ := E₁) (M₁' := E)
        (g.compContinuousLinearMap (M₁ := E) (M₁' := E₁) fun (i : ι) ↦ (↑(↑(f i) : E i ≃L[𝕜] E₁ i) : E i →L[𝕜] E₁ i))
        fun (i : ι) ↦ (↑(↑(f i).symm : E₁ i ≃L[𝕜] E i) : E₁ i →L[𝕜] E i)) ≤
    norm (E := ContinuousMultilinearMap 𝕜 E G)
      (g.compContinuousLinearMap (M₁ := E) (M₁' := E₁) fun (i : ι) ↦ (f i).toLinearIsometry.toContinuousLinearMap)

Tactic:
apply (g.compContinuousLinearMap fun i =>
    (f i : E i →L[𝕜] E₁ i)).norm_compContinuous_linearIsometry_le
      fun i => (f i).symm.toLinearIsometry

Premises used: LinearIsometryEquiv.toLinearIsometry, LinearIsometryEquiv.symm, ContinuousMultilinearMap.norm_compContinuous_linearIsometry_le, LinearIsometryEquiv.toContinuousLinearEquiv, ContinuousMultilinearMap.compContinuousLinearMap, ContinuousLinearEquiv.toContinuousLinearMap
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst : NontriviallyNormedField 𝕜
inst_1 : (i : ι) → SeminormedAddCommGroup (E i)
inst_2 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_3 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_4 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_5 : SeminormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
inst_7 : Fintype ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
⊢ LE.le (α := ℝ) 0 (∏ i : ι, ‖f i‖)

Tactic:
positivity
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
H :
  ∀ (x : (i : ι) → E i),
    Eq.{wG + 1} (α := G)
      ((g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
              (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                (f₁ + f₂)) :
            (a : (i : ι) → E i) → G)
          x :
        G)
      ((g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
                (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₁) +
              g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
                (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₂) :
            (a : (i : ι) → E i) → G)
          x :
        G)
⊢ g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
      (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i (f₁ + f₂)) =
    g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
        (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₁) +
      g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
        (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₂)

Tactic:
ext x
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
x : (i : ι) → E i
H :
  ((g : (a : (i : ι) → E₁ i) → G) fun (i_1 : ι) ↦
      (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i (f₁ + f₂) i_1 :
          (a : E i_1) → E₁ i_1)
        (x i_1)) =
    HAdd.hAdd (α := G) (β := G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (i_1 : ι) ↦
          (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₁ i_1 :
              (a : E i_1) → E₁ i_1)
            (x i_1) :
        G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (i_1 : ι) ↦
          (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₂ i_1 :
              (a : E i_1) → E₁ i_1)
            (x i_1) :
        G)
⊢ Eq.{wG + 1} (α := G)
    ((g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
            (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
              (f₁ + f₂)) :
          (a : (i : ι) → E i) → G)
        x :
      G)
    ((g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
              (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₁) +
            g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
              (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₂) :
          (a : (i : ι) → E i) → G)
        x :
      G)

Tactic:
simp only [compContinuousLinearMap_apply, add_apply]

Premises used: ContinuousMultilinearMap.compContinuousLinearMap_apply, ContinuousMultilinearMap.add_apply
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
x : (i : ι) → E i
⊢ (g : (a : (i : ι) → E₁ i) → G)
      (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i
        (HAdd.hAdd (α := E₁ i) (β := E₁ i) ((f₁ : (a : E i) → E₁ i) (x i) : E₁ i)
          ((f₂ : (a : E i) → E₁ i) (x i) : E₁ i))) =
    HAdd.hAdd (α := G) (β := G)
      ((g : (a : (i : ι) → E₁ i) → G)
          (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₁ : (a : E i) → E₁ i) (x i))) :
        G)
      ((g : (a : (i : ι) → E₁ i) → G)
          (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₂ : (a : E i) → E₁ i) (x i))) :
        G)

Tactic:
g.map_update_add (fun j ↦ f j (x j)) i (f₁ (x i)) (f₂ (x i))

Premises used: ContinuousMultilinearMap.map_update_add, DFunLike.coe
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
x : (i : ι) → E i
h₁ :
  ∀ (x_1 : ι),
    (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i (f₁ + f₂) x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) =
      update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i
        (HAdd.hAdd (α := E₁ i) (β := E₁ i) ((f₁ : (a : E i) → E₁ i) (x i) : E₁ i)
          ((f₂ : (a : E i) → E₁ i) (x i) : E₁ i))
        x_1
h₂ :
  ∀ (x_1 : ι),
    Eq.{wE₁ + 1} (α := E₁ x_1)
      ((update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₁ x_1 :
            (a : E x_1) → E₁ x_1)
          (x x_1) :
        E₁ x_1)
      (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₁ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)
h₃ :
  ∀ (x_1 : ι),
    Eq.{wE₁ + 1} (α := E₁ x_1)
      ((update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₂ x_1 :
            (a : E x_1) → E₁ x_1)
          (x x_1) :
        E₁ x_1)
      (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₂ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)
⊢ ((g : (a : (i : ι) → E₁ i) → G) fun (i_1 : ι) ↦
      (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i (f₁ + f₂) i_1 :
          (a : E i_1) → E₁ i_1)
        (x i_1)) =
    HAdd.hAdd (α := G) (β := G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (i_1 : ι) ↦
          (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₁ i_1 :
              (a : E i_1) → E₁ i_1)
            (x i_1) :
        G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (i_1 : ι) ↦
          (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₂ i_1 :
              (a : E i_1) → E₁ i_1)
            (x i_1) :
        G)

Tactic:
convert! g.map_update_add (fun j ↦ f j (x j)) i (f₁ (x i)) (f₂ (x i))

Premises used: ContinuousMultilinearMap.map_update_add, DFunLike.coe
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
x : (i : ι) → E i
x_1 : ι
⊢ (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i (f₁ + f₂) x_1 :
        (a : E x_1) → E₁ x_1)
      (x x_1) =
    update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i
      (HAdd.hAdd (α := E₁ i) (β := E₁ i) ((f₁ : (a : E i) → E₁ i) (x i) : E₁ i) ((f₂ : (a : E i) → E₁ i) (x i) : E₁ i))
      x_1

Tactic:
apply_update (fun (i : ι) (f : E i →L[𝕜] E₁ i) ↦ f (x i)) f i _ _

Premises used: Function.apply_update, DFunLike.coe
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
x : (i : ι) → E i
x_1 : ι
⊢ (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i (f₁ + f₂) x_1 :
        (a : E x_1) → E₁ x_1)
      (x x_1) =
    update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i
      (HAdd.hAdd (α := E₁ i) (β := E₁ i) ((f₁ : (a : E i) → E₁ i) (x i) : E₁ i) ((f₂ : (a : E i) → E₁ i) (x i) : E₁ i))
      x_1

Tactic:
exact apply_update (fun (i : ι) (f : E i →L[𝕜] E₁ i) ↦ f (x i)) f i _ _

Premises used: Function.apply_update, DFunLike.coe
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
x : (i : ι) → E i
x_1 : ι
⊢ Eq.{wE₁ + 1} (α := E₁ x_1)
    ((update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₁ x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) :
      E₁ x_1)
    (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₁ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)

Tactic:
apply_update (fun (i : ι) (f : E i →L[𝕜] E₁ i) ↦ f (x i)) f i _ _

Premises used: Function.apply_update, DFunLike.coe
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
x : (i : ι) → E i
x_1 : ι
⊢ Eq.{wE₁ + 1} (α := E₁ x_1)
    ((update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₁ x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) :
      E₁ x_1)
    (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₁ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)

Tactic:
exact apply_update (fun (i : ι) (f : E i →L[𝕜] E₁ i) ↦ f (x i)) f i _ _

Premises used: Function.apply_update, DFunLike.coe
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
x : (i : ι) → E i
x_1 : ι
⊢ Eq.{wE₁ + 1} (α := E₁ x_1)
    ((update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₂ x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) :
      E₁ x_1)
    (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₂ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)

Tactic:
apply_update (fun (i : ι) (f : E i →L[𝕜] E₁ i) ↦ f (x i)) f i _ _

Premises used: Function.apply_update, DFunLike.coe
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
x : (i : ι) → E i
x_1 : ι
⊢ Eq.{wE₁ + 1} (α := E₁ x_1)
    ((update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₂ x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) :
      E₁ x_1)
    (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₂ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)

Tactic:
exact apply_update (fun (i : ι) (f : E i →L[𝕜] E₁ i) ↦ f (x i)) f i _ _

Premises used: Function.apply_update, DFunLike.coe
-- Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
a : 𝕜
f₀ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
H :
  ∀ (x : (i : ι) → E i),
    Eq.{wG + 1} (α := G)
      ((g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
              (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                (a • f₀)) :
            (a : (i : ι) → E i) → G)
          x :
        G)
      ((a •
              g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
                (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₀) :
            (a : (i : ι) → E i) → G)
          x :
        G)
⊢ g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
      (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i (a • f₀)) =
    a •
      g.compContinuousLinearMap (M₁ := E) (M₁' := E₁)
        (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₀)

Tactic:
ext x