/-- `(φ i ⋆ g i) (k i)` tends to `z₀` as `i` tends to some filter `l` if
* `φ` is a sequence of nonnegative functions with integral `1` as `i` tends to `l`;
* The support of `φ` tends to small neighborhoods around `(0 : G)` as `i` tends to `l`;
* `g i` is `mu`-a.e. strongly measurable as `i` tends to `l`;
* `g i x` tends to `z₀` as `(i, x)` tends to `l ×ˢ 𝓝 x₀`;
* `k i` tends to `x₀`.

See also `ContDiffBump.convolution_tendsto_right`.
-/
theorem convolution_tendsto_right {ι} {g : ι → G → E'} {l : Filter ι} {x₀ : G} {z₀ : E'}
    {φ : ι → G → ℝ} {k : ι → G} (hnφ : ∀ᶠ i in l, ∀ x, 0 ≤ φ i x)
    (hiφ : ∀ᶠ i in l, ∫ x, φ i x ∂μ = 1)
    -- todo: we could weaken this to "the integral tends to 1"
    (hφ : Tendsto (fun n => support (φ n)) l (𝓝 0).smallSets)
    (hmg : ∀ᶠ i in l, AEStronglyMeasurable (g i) μ) (hcg : Tendsto (uncurry g) (l ×ˢ 𝓝 x₀) (𝓝 z₀))
    (hk : Tendsto k l (𝓝 x₀)) :
    Tendsto (fun i : ι => (φ i ⋆[lsmul ℝ ℝ, μ] g i : G → E') (k i)) l (𝓝 z₀) := by
  simp_rw [tendsto_smallSets_iff] at hφ
  rw [Metric.tendsto_nhds] at hcg ⊢
  simp_rw [Metric.eventually_prod_nhds_iff] at hcg
  intro ε hε
  have h2ε : 0 < ε / 3 := div_pos hε (by norm_num)
  obtain ⟨p, hp, δ, hδ, hgδ⟩ := hcg _ h2ε
  dsimp only [uncurry] at hgδ
  have h2k := hk.eventually (ball_mem_nhds x₀ <| half_pos hδ)
  have h2φ := hφ (ball (0 : G) _) <| ball_mem_nhds _ (half_pos hδ)
  filter_upwards [hp, h2k, h2φ, hnφ, hiφ, hmg] with i hpi hki hφi hnφi hiφi hmgi
  have hgi : dist (g i (k i)) z₀ < ε / 3 := hgδ hpi (hki.trans <| half_lt_self hδ)
  have h1 : ∀ x' ∈ ball (k i) (δ / 2), dist (g i x') (g i (k i)) ≤ ε / 3 + ε / 3 := by
    intro x' hx'
    refine' (dist_triangle_right _ _ _).trans (add_le_add (hgδ hpi _).le hgi.le)
    exact ((dist_triangle _ _ _).trans_lt (add_lt_add hx'.out hki)).trans_eq (add_halves δ)
  

Proof state:
case h
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝¹⁸ : NormedAddCommGroup E
inst✝¹⁷ : NormedAddCommGroup E'
inst✝¹⁶ : NormedAddCommGroup E''
inst✝¹⁵ : NormedAddCommGroup F
f f' : G → E
g✝ g' : G → E'
x x' : G
y y' : E
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : NormedSpace 𝕜 E
inst✝¹² : NormedSpace 𝕜 E'
inst✝¹¹ : NormedSpace 𝕜 E''
inst✝¹⁰ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁹ : MeasurableSpace G
μ ν : Measure G
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : SeminormedAddCommGroup G
inst✝⁶ : BorelSpace G
inst✝⁵ : SecondCountableTopology G
inst✝⁴ : IsAddLeftInvariant μ
inst✝³ : SigmaFinite μ
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ E'
inst✝ : CompleteSpace E'
ι : Type u_1
g : ι → G → E'
l : Filter ι
x₀ : G
z₀ : E'
φ : ι → G → ℝ
k : ι → G
hnφ : ∀ᶠ (i : ι) in l, ∀ (x : G), 0 ≤ φ i x
hiφ : ∀ᶠ (i : ι) in l, ∫ (x : G), φ i x ∂μ = 1
hmg : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (g i) μ
hk : Tendsto k l (𝓝 x₀)
hφ : ∀ t ∈ 𝓝 0, ∀ᶠ (x : ι) in l, support (φ x) ⊆ t
hcg :
  ∀ ε > 0,
    ∃ pa,
      (∀ᶠ (i : ι) in l, pa i) ∧ ∃ ε_1 > 0, ∀ {i : ι}, pa i → ∀ {x : G}, dist x x₀ < ε_1 → dist (uncurry g (i, x)) z₀ < ε
ε : ℝ
hε : ε > 0
h2ε : 0 < ε / 3
p : ι → Prop
hp : ∀ᶠ (i : ι) in l, p i
δ : ℝ
hδ : δ > 0
hgδ : ∀ {i : ι}, p i → ∀ {x : G}, dist x x₀ < δ → dist (g i x) z₀ < ε / 3
h2k : ∀ᶠ (x : ι) in l, dist (k x) x₀ < δ / 2
h2φ : ∀ᶠ (x : ι) in l, support (φ x) ⊆ ball 0 (δ / 2)
i : ι
hpi : p i
hki : dist (k i) x₀ < δ / 2
hφi : support (φ i) ⊆ ball 0 (δ / 2)
hnφi : ∀ (x : G), 0 ≤ φ i x
hiφi : ∫ (x : G), φ i x ∂μ = 1
hmgi : AEStronglyMeasurable (g i) μ
hgi : dist (g i (k i)) z₀ < ε / 3
h1 : ∀ x' ∈ ball (k i) (δ / 2), dist (g i x') (g i (k i)) ≤ ε / 3 + ε / 3
⊢ dist (φ i ⋆[lsmul ℝ ℝ, k i] g i) z₀ < ε

Next tactic:
have := dist_convolution_le (add_pos h2ε h2ε).le hφi hnφi hiφi hmgi h1