Theorem: isBoundedLinearMap_continuousMultilinearMap_comp_linear
File: Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
ι : Type u_5
inst✝ : Fintype ι
g : G →L[𝕜] E
⊢ IsBoundedLinearMap 𝕜 fun f => f.compContinuousLinearMap fun x => g
state_after: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
ι : Type u_5
inst✝ : Fintype ι
g : G →L[𝕜] E
f : ContinuousMultilinearMap 𝕜 (fun x => E) F
⊢ ‖f.compContinuousLinearMap fun x => g‖ ≤ ‖g‖ ^ Fintype.card ι * ‖f‖
tactic: refine
    IsLinearMap.with_bound
      ⟨fun f₁ f₂ => by ext; rfl,
        fun c f => by ext; rfl⟩
      (‖g‖ ^ Fintype.card ι) fun f => ?_
state_before: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
ι : Type u_5
inst✝ : Fintype ι
g : G →L[𝕜] E
f : ContinuousMultilinearMap 𝕜 (fun x => E) F
⊢ ‖f.compContinuousLinearMap fun x => g‖ ≤ ‖g‖ ^ Fintype.card ι * ‖f‖
state_after: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
ι : Type u_5
inst✝ : Fintype ι
g : G →L[𝕜] E
f : ContinuousMultilinearMap 𝕜 (fun x => E) F
⊢ 0 ≤ ‖g‖ ^ Fintype.card ι * ‖f‖

𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
ι : Type u_5
inst✝ : Fintype ι
g : G →L[𝕜] E
f : ContinuousMultilinearMap 𝕜 (fun x => E) F
⊢ ∀ (m : ι → G), ‖(f.compContinuousLinearMap fun x => g) m‖ ≤ ‖g‖ ^ Fintype.card ι * ‖f‖ * ∏ i : ι, ‖m i‖
tactic: apply ContinuousMultilinearMap.opNorm_le_bound _ _ _
state_before: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
ι : Type u_5
inst✝ : Fintype ι
g : G →L[𝕜] E
f : ContinuousMultilinearMap 𝕜 (fun x => E) F
⊢ ∀ (m : ι → G), ‖(f.compContinuousLinearMap fun x => g) m‖ ≤ ‖g‖ ^ Fintype.card ι * ‖f‖ * ∏ i : ι, ‖m i‖
state_after: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
ι : Type u_5
inst✝ : Fintype ι
g : G →L[𝕜] E
f : ContinuousMultilinearMap 𝕜 (fun x => E) F
m : ι → G
⊢ ‖(f.compContinuousLinearMap fun x => g) m‖ ≤ ‖g‖ ^ Fintype.card ι * ‖f‖ * ∏ i : ι, ‖m i‖
tactic: intro m
state_before: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
ι : Type u_5
inst✝ : Fintype ι
g : G →L[𝕜] E
f : ContinuousMultilinearMap 𝕜 (fun x => E) F
m : ι → G
⊢ ‖(f.compContinuousLinearMap fun x => g) m‖ ≤ ‖g‖ ^ Fintype.card ι * ‖f‖ * ∏ i : ι, ‖m i‖
state_after: no goals
tactic: calc
    ‖f (g ∘ m)‖ ≤ ‖f‖ * ∏ i, ‖g (m i)‖ := f.le_opNorm _
    _ ≤ ‖f‖ * ∏ i, ‖g‖ * ‖m i‖ := by
      apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
      exact Finset.prod_le_prod (fun i _ => norm_nonneg _) fun i _ => g.le_opNorm _
    _ = ‖g‖ ^ Fintype.card ι * ‖f‖ * ∏ i, ‖m i‖ := by
      simp only [Finset.prod_mul_distrib, Finset.prod_const, Finset.card_univ]
      ring