-- Module: Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm
Goal:
ι : Type uι
inst : Fintype ι
𝕜 : Type u𝕜
inst_1 : NontriviallyNormedField 𝕜
E : ι → Type uE
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u𝕜, uE} 𝕜 (E i)
F : Type uF
inst_4 : SeminormedAddCommGroup F
inst_5 : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x_1 : ⨂[𝕜] (i : ι), E i
G : Type (max (max uE u𝕜) uι) :=
  HasQuotient.Quotient (⨂[𝕜] (i : ι), E i) (B := Submodule 𝕜 (⨂[𝕜] (i : ι), E i))
    (LinearMap.ker (R₂ := 𝕜) (M₂ := F) (τ₁₂ := RingHom.id 𝕜)
      ((lift (s := E) : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) (f.toMultilinearMap (M₁ := E))))
G' : Submodule 𝕜 F :=
  LinearMap.range (R := 𝕜) (M := ⨂[𝕜] (i : ι), E i) (τ₁₂ := RingHom.id 𝕜)
    ((lift (s := E) : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) (f.toMultilinearMap (M₁ := E)))
e : HasQuotient.Quotient (⨂[𝕜] (i : ι), E i) (B := Submodule 𝕜 (⨂[𝕜] (i : ι), E i))
    (LinearMap.ker (R₂ := 𝕜) (M₂ := F) (τ₁₂ := RingHom.id 𝕜)
      ((lift (s := E) : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) (f.toMultilinearMap (M₁ := E)))) ≃ₗ[𝕜]
  ↥(LinearMap.range (R := 𝕜) (M := ⨂[𝕜] (i : ι), E i) (τ₁₂ := RingHom.id 𝕜)
      ((lift (s := E) : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) (f.toMultilinearMap (M₁ := E)))) :=
  LinearMap.quotKerEquivRange
    ((lift (s := E) : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) (f.toMultilinearMap (M₁ := E)))
this_1 : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E
  (HasQuotient.Quotient (⨂[𝕜] (i : ι), E i) (B := Submodule 𝕜 (⨂[𝕜] (i : ι), E i))
    (LinearMap.ker (R₂ := 𝕜) (M₂ := F) (τ₁₂ := RingHom.id 𝕜)
      ((lift (s := E) : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) (f.toMultilinearMap (M₁ := E))))) :=
  (LinearEquiv.symm (lift (s := E)) :
      (a :
          (⨂[𝕜] (i : ι), E i) →ₗ[𝕜]
            (⨂[𝕜] (i : ι), E i) ⧸
              ((lift : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) f.toMultilinearMap).ker) →
        MultilinearMap 𝕜 E
          ((⨂[𝕜] (i : ι), E i) ⧸
            ((lift : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) f.toMultilinearMap).ker))
    ((↑e.symm :
        ↥((lift : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) f.toMultilinearMap).range →ₗ[𝕜]
          (⨂[𝕜] (i : ι), E i) ⧸
            ((lift : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) f.toMultilinearMap).ker) ∘ₗ
      LinearMap.rangeRestrict
        ((lift (s := E) : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) (f.toMultilinearMap (M₁ := E))))
x : (i : ι) → E i
simp : norm.{uF} (E := F) ((f : (a : (i : ι) → E i) → F) x : F) ≤ ‖f‖ * ∏ i : ι, ‖x i‖
⊢ norm.{uF} (E :=
      ↥(LinearMap.range (R := 𝕜) (M := ⨂[𝕜] (i : ι), E i) (τ₁₂ := RingHom.id 𝕜)
          ((lift (s := E) : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) (f.toMultilinearMap (M₁ := E)))))
      ((e :
            (a :
                (⨂[𝕜] (i : ι), E i) ⧸
                  ((lift : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) f.toMultilinearMap).ker) →
              ↥((lift : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) f.toMultilinearMap).range)
          ((f'₀ :
              (a : (i : ι) → E i) →
                (⨂[𝕜] (i : ι), E i) ⧸
                  ((lift : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F) f.toMultilinearMap).ker)
            x) :
        ↥(LinearMap.range (R := 𝕜) (M := ⨂[𝕜] (i : ι), E i) (τ₁₂ := RingHom.id 𝕜)
            ((lift (s := E) : (a : MultilinearMap 𝕜 E F) → (⨂[𝕜] (i : ι), E i) →ₗ[𝕜] F)
              (f.toMultilinearMap (M₁ := E))))) ≤
    ‖f‖ * ∏ i : ι, ‖x i‖

Tactic:
simp only [lift_symm, LinearMap.compMultilinearMap_apply, LinearMap.coe_comp,
        LinearEquiv.coe_coe, Function.comp_apply, LinearEquiv.apply_symm_apply, Submodule.coe_norm,
        LinearMap.codRestrict_apply, lift.tprod, ContinuousMultilinearMap.coe_coe, e, f'₀]

Premises used: PiTensorProduct.lift.tprod, ContinuousMultilinearMap.coe_coe, LinearMap.coe_comp, LinearEquiv.apply_symm_apply, Function.comp_apply, Submodule.coe_norm, LinearMap.codRestrict_apply, LinearMap.compMultilinearMap_apply, LinearEquiv.coe_coe, PiTensorProduct.lift_symm