Theorem: PiTensorProduct.dualSeminorms_bounded
File: Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean

Proof steps:
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ BddAbove
    {p |
      ∃ G x x_1, p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)}
state_after: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ projectiveSeminorm ∈
    upperBounds
      {p |
        ∃ G x x_1,
          p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)}
tactic: existsi projectiveSeminorm
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ projectiveSeminorm ∈
    upperBounds
      {p |
        ∃ G x x_1,
          p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)}
state_after: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ ∀
    x ∈
      {p |
        ∃ G x x_1,
          p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)},
    x ≤ projectiveSeminorm
tactic: rw [mem_upperBounds]
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ ∀
    x ∈
      {p |
        ∃ G x x_1,
          p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)},
    x ≤ projectiveSeminorm
state_after: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ ∀ (x : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)) (x_1 : Type (max uι u𝕜 uE)) (x_2 : SeminormedAddCommGroup x_1)
    (x_3 : NormedSpace 𝕜 x_1),
    x = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E x_1 →L[𝕜] x_1)).comp (toDualContinuousMultilinearMap x_1) →
      x ≤ projectiveSeminorm
tactic: simp only [Set.mem_setOf_eq, forall_exists_index]
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ ∀ (x : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)) (x_1 : Type (max uι u𝕜 uE)) (x_2 : SeminormedAddCommGroup x_1)
    (x_3 : NormedSpace 𝕜 x_1),
    x = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E x_1 →L[𝕜] x_1)).comp (toDualContinuousMultilinearMap x_1) →
      x ≤ projectiveSeminorm
state_after: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
x✝¹ : SeminormedAddCommGroup G
x✝ : NormedSpace 𝕜 G
hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
⊢ p ≤ projectiveSeminorm
tactic: intro p G _ _ hp
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
x✝¹ : SeminormedAddCommGroup G
x✝ : NormedSpace 𝕜 G
hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
⊢ p ≤ projectiveSeminorm
state_after: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
x✝¹ : SeminormedAddCommGroup G
x✝ : NormedSpace 𝕜 G
hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
⊢ (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G) ≤ projectiveSeminorm
tactic: rw [hp]
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
x✝¹ : SeminormedAddCommGroup G
x✝ : NormedSpace 𝕜 G
hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
⊢ (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G) ≤ projectiveSeminorm
state_after: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
x✝¹ : SeminormedAddCommGroup G
x✝ : NormedSpace 𝕜 G
hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
x : ⨂[𝕜] (i : ι), E i
⊢ (fun f => ⇑f) ((normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)) x ≤
    (fun f => ⇑f) projectiveSeminorm x
tactic: intro x
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
x✝¹ : SeminormedAddCommGroup G
x✝ : NormedSpace 𝕜 G
hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
x : ⨂[𝕜] (i : ι), E i
⊢ (fun f => ⇑f) ((normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)) x ≤
    (fun f => ⇑f) projectiveSeminorm x
state_after: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
x✝¹ : SeminormedAddCommGroup G
x✝ : NormedSpace 𝕜 G
hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
x : ⨂[𝕜] (i : ι), E i
⊢ ‖(toDualContinuousMultilinearMap G) x‖ ≤ projectiveSeminorm x
tactic: simp only [Seminorm.comp_apply, coe_normSeminorm]
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
x✝¹ : SeminormedAddCommGroup G
x✝ : NormedSpace 𝕜 G
hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
x : ⨂[𝕜] (i : ι), E i
⊢ ‖(toDualContinuousMultilinearMap G) x‖ ≤ projectiveSeminorm x
state_after: no goals
tactic: exact toDualContinuousMultilinearMap_le_projectiveSeminorm _