/-- The map `u ↦ f • u` is an isometry between the `L^1` spaces for `μ.withDensity f` and `μ`. -/
noncomputable def withDensitySMulLI {f : α → ℝ≥0} (f_meas : Measurable f) :
    Lp E 1 (μ.withDensity fun x => f x) →ₗᵢ[ℝ] Lp E 1 μ where
  toFun u := (memℒ1_smul_of_L1_withDensity f_meas u).toLp _
  map_add' := by
    intro u v
    ext1
    filter_upwards [(memℒ1_smul_of_L1_withDensity f_meas u).coeFn_toLp,
      (memℒ1_smul_of_L1_withDensity f_meas v).coeFn_toLp,
      (memℒ1_smul_of_L1_withDensity f_meas (u + v)).coeFn_toLp,
      Lp.coeFn_add ((memℒ1_smul_of_L1_withDensity f_meas u).toLp _)
        ((memℒ1_smul_of_L1_withDensity f_meas v).toLp _),
      (ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 (Lp.coeFn_add u v)]
    intro x hu hv huv h' h''
    rw [huv, h', Pi.add_apply, hu, hv]
    rcases eq_or_ne (f x) 0 with (hx | hx)
    · simp only [hx, zero_smul, add_zero]
    · rw [h'' _, Pi.add_apply, smul_add]
      simpa only [Ne.def, ENNReal.coe_eq_zero] using hx
  map_smul' := by
    intro r u
    ext1
    filter_upwards [(ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 (Lp.coeFn_smul r u),
      (memℒ1_smul_of_L1_withDensity f_meas (r • u)).coeFn_toLp,
      Lp.coeFn_smul r ((memℒ1_smul_of_L1_withDensity f_meas u).toLp _),
      (memℒ1_smul_of_L1_withDensity f_meas u).coeFn_toLp]
    intro x h h' h'' h'''
    rw [RingHom.id_apply, h', h'', Pi.smul_apply, h''']
    rcases eq_or_ne (f x) 0 with (hx | hx)
    · simp only [hx, zero_smul, smul_zero]
    · rw [h _, smul_comm, Pi.smul_apply]
      simpa only [Ne.def, ENNReal.coe_eq_zero] using hx
  norm_map' := by
    intro u
    -- Porting note: Lean can't infer types of `AddHom.coe_mk`.
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : MeasurableSpace δ
inst✝³ : NormedAddCommGroup β
inst✝² : NormedAddCommGroup γ
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
u : ↥(Lp E 1)
⊢ ‖{
          toAddHom :=
            { toFun := fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1),
              map_add' :=
                (_ :
                  ∀ (u v : ↥(Lp E 1)),
                    (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) (u + v) =
                      (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) u +
                        (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) v) },
          map_smul' :=
            (_ :
              ∀ (r : ℝ) (u : ↥(Lp E 1)),
                AddHom.toFun
                    { toFun := fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1),
                      map_add' :=
                        (_ :
                          ∀ (u v : ↥(Lp E 1)),
                            (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) (u + v) =
                              (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) u +
                                (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) v) }
                    (r • u) =
                  (RingHom.id ℝ) r •
                    AddHom.toFun
                      { toFun := fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1),
                        map_add' :=
                          (_ :
                            ∀ (u v : ↥(Lp E 1)),
                              (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1))
                                  (u + v) =
                                (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) u +
                                  (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) v) }
                      u) }
        u‖ =
    ‖u‖

Next tactic:
simp only [snorm, LinearMap.coe_mk,
      AddHom.coe_mk (M := Lp E 1 (μ.withDensity fun x => f x)) (N := Lp E 1 μ), Lp.norm_toLp,
      one_ne_zero, ENNReal.one_ne_top, ENNReal.one_toReal, if_false, snorm', ENNReal.rpow_one,
      _root_.div_one, Lp.norm_def]