-- Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean
theorem setLIntegral_nnnorm_condExpIndSMul_le (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (x : G) {t : Set α} (ht : MeasurableSet[m] t) (hμt : μ t ≠ ∞) :
    (∫⁻ a in t, ‖condExpIndSMul hm hs hμs x a‖₊ ∂μ) ≤ μ (s ∩ t) * ‖x‖₊ :=
  calc
    ∫⁻ a in t, ‖condExpIndSMul hm hs hμs x a‖₊ ∂μ =
        ∫⁻ a in t, ‖(condExpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a • x‖₊ ∂μ :=
      setLIntegral_congr_fun (hm t ht)
        ((condExpIndSMul_ae_eq_smul hm hs hμs x).mono fun a ha _ => by rw [ha])
    _ = (∫⁻ a in t, ‖(condExpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a‖₊ ∂μ) * ‖x‖₊

Proof state:
case hf
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ Measurable fun a => ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊

Tactic:
exact (Lp.stronglyMeasurable _).enorm

Result:
no goals

Proof state:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ ∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x‖₊ ∂μ =
    (∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ ∂μ) * ↑‖x‖₊

Tactic:
simp_rw [nnnorm_smul, ENNReal.coe_mul]

Result:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ ∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ * ↑‖x‖₊ ∂μ =
    (∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ ∂μ) * ↑‖x‖₊

Proof state:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ ∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ * ↑‖x‖₊ ∂μ =
    (∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ ∂μ) * ↑‖x‖₊

Tactic:
rw [lintegral_mul_const]

Result:
case hf
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ Measurable fun a => ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊

Proof state:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
a : α
ha : ↑↑(condExpIndSMul hm hs hμs x) a = (fun a => ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) a
x✝ : a ∈ t
⊢ ↑‖↑↑(condExpIndSMul hm hs hμs x) a‖₊ = ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x‖₊

Tactic:
rw [ha]

Result:
no goals