-- Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
lemma range_toUniformOnFun [DecidableEq ι] [TopologicalSpace F] :
    range toUniformOnFun =
      {f : (Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F |
        Continuous (toFun _ f) ∧
        (∀ (m : Π i, E i) i x y,
          toFun _ f (update m i (x + y)) = toFun _ f (update m i x) + toFun _ f (update m i y)) ∧
        (∀ (m : Π i, E i) i (c : 𝕜) x,
          toFun _ f (update m i (c • x)) = c • toFun _ f (update m i x))}

Proof state:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
⊢ range toUniformOnFun =
    {f |
      Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
        (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
            (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
          ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
            (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
              c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)}

Tactic:
ext f

Result:
case h
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
⊢ f ∈ range toUniformOnFun ↔
    f ∈
      {f |
        Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
          (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
                (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
            ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
                c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)}

Proof state:
case h
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
⊢ f ∈ range toUniformOnFun ↔
    f ∈
      {f |
        Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
          (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
                (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
            ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
                c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)}

Tactic:
constructor

Result:
case h.mp
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
⊢ f ∈ range toUniformOnFun →
    f ∈
      {f |
        Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
          (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
                (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
            ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
                c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)}

case h.mpr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
⊢ f ∈
      {f |
        Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
          (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
                (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
            ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
                c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)} →
    f ∈ range toUniformOnFun

Proof state:
case h.mp.intro
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
f : ContinuousMultilinearMap 𝕜 E F
⊢ f.toUniformOnFun ∈
    {f |
      Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
        (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
            (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
          ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
            (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
              c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)}

Tactic:
exact ⟨f.cont, f.map_update_add, f.map_update_smul⟩

Result:
no goals

Proof state:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁸ : NormedField 𝕜
inst✝⁷ : (i : ι) → TopologicalSpace (E i)
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : DecidableEq ι
inst✝¹ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
hcont : Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f)
hadd :
  ∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
    (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
      (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)
hsmul :
  ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
    (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) = c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)
inst✝ : DecidableEq ι
⊢ ∀ (m : (i : ι) → E i) (i : ι) (x y : E i), f (update m i (x + y)) = f (update m i x) + f (update m i y)

Tactic:
convert hadd

Result:
no goals

Proof state:
case h.mpr.intro.intro
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
hcont : Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f)
hadd :
  ∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
    (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
      (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)
hsmul :
  ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
    (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) = c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)
⊢ f ∈ range toUniformOnFun

Tactic:
exact ⟨⟨⟨f, by intro; convert hadd, by intro; convert hsmul⟩, hcont⟩, rfl⟩

Result:
no goals