-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
theorem injectiveSeminorm_le_projectiveSeminorm :
    injectiveSeminorm (𝕜 := 𝕜) (E := E) ≤ projectiveSeminorm

Proof state:
case refine_1
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ {p |
      ∃ G x x_1,
        p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)}.Nonempty

Tactic:
existsi 0

Result:
case refine_1
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ 0 ∈
    {p |
      ∃ G x x_1, p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)}

Proof state:
case refine_1
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ ∃ G x x_1, 0 = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)

Tactic:
existsi PUnit, inferInstance, inferInstance

Result:
case refine_1
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ 0 =
    (normSeminorm 𝕜
          (ContinuousMultilinearMap 𝕜 E PUnit.{max (max (uE + 1) (uι + 1)) (u𝕜 + 1)} →L[𝕜]
            PUnit.{max (max (uE + 1) (uι + 1)) (u𝕜 + 1)})).comp
      (toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (uι + 1)) (u𝕜 + 1)})

Proof state:
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ injectiveSeminorm ≤ projectiveSeminorm

Tactic:
rw [injectiveSeminorm]

Result:
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ sSup
      {p |
        ∃ G x x_1,
          p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)} ≤
    projectiveSeminorm

Proof state:
case refine_2
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
hp :
  p ∈
    {p |
      ∃ G x x_1, p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)}
⊢ p ≤ projectiveSeminorm

Tactic:
simp only [Set.mem_setOf_eq] at hp

Result:
case refine_2
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
hp : ∃ G x x_1, p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
⊢ p ≤ projectiveSeminorm

Proof state:
case refine_2.intro.intro.intro
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
w✝¹ : SeminormedAddCommGroup G
w✝ : NormedSpace 𝕜 G
h : 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:
simp only [Seminorm.comp_apply, coe_normSeminorm]

Result:
case refine_2.intro.intro.intro
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
w✝¹ : SeminormedAddCommGroup G
w✝ : NormedSpace 𝕜 G
h : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
x : ⨂[𝕜] (i : ι), E i
⊢ ‖(toDualContinuousMultilinearMap G) x‖ ≤ projectiveSeminorm x

Proof state:
case refine_1
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ 0 =
    (normSeminorm 𝕜
          (ContinuousMultilinearMap 𝕜 E PUnit.{max (max (uE + 1) (uι + 1)) (u𝕜 + 1)} →L[𝕜]
            PUnit.{max (max (uE + 1) (uι + 1)) (u𝕜 + 1)})).comp
      (toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (uι + 1)) (u𝕜 + 1)})

Tactic:
ext x

Result:
case refine_1.h
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
x : ⨂[𝕜] (i : ι), E i
⊢ 0 x =
    ((normSeminorm 𝕜
            (ContinuousMultilinearMap 𝕜 E PUnit.{max (max (uE + 1) (uι + 1)) (u𝕜 + 1)} →L[𝕜]
              PUnit.{max (max (uE + 1) (uι + 1)) (u𝕜 + 1)})).comp
        (toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (uι + 1)) (u𝕜 + 1)}))
      x