l3lab/ntp-mathlib huggingface dataset:
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
  have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ :=
    integral_nonneg fun a => norm_nonneg _
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
    ENNReal.toReal_ofReal this,
    ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_nnnorm]
  swap; · 

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
this : 0 ≤ ∫ (a : α), ‖↑↑(condexpIndL1Fin hm hs hμs x) a‖ ∂μ
⊢ Integrable fun a => ↑↑(condexpIndL1Fin hm hs hμs x) a

Next tactic:
rw [← memℒp_one_iff_integrable]
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
  have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ :=
    integral_nonneg fun a => norm_nonneg _
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
    ENNReal.toReal_ofReal this,
    ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_nnnorm]
  swap; · rw [← memℒp_one_iff_integrable]; 

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
this : 0 ≤ ∫ (a : α), ‖↑↑(condexpIndL1Fin hm hs hμs x) a‖ ∂μ
⊢ Memℒp (fun a => ↑↑(condexpIndL1Fin hm hs hμs x) a) 1

Next tactic:
exact Lp.memℒp _
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
  have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ :=
    integral_nonneg fun a => norm_nonneg _
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
    ENNReal.toReal_ofReal this,
    ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_nnnorm]
  swap; · rw [← memℒp_one_iff_integrable]; exact Lp.memℒp _
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
this : 0 ≤ ∫ (a : α), ‖↑↑(condexpIndL1Fin hm hs hμs x) a‖ ∂μ
⊢ ∫⁻ (x_1 : α), ↑‖↑↑(condexpIndL1Fin hm hs hμs x) x_1‖₊ ∂μ ≤ ↑↑μ s * ENNReal.ofReal ‖x‖

Next tactic:
have h_eq :
    ∫⁻ a, ‖condexpIndL1Fin hm hs hμs x a‖₊ ∂μ = ∫⁻ a, ‖condexpIndSMul hm hs hμs x a‖₊ ∂μ := by
    refine' lintegral_congr_ae _
    refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun z hz => _
    dsimp only
    rw [hz]
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
  have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ :=
    integral_nonneg fun a => norm_nonneg _
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
    ENNReal.toReal_ofReal this,
    ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_nnnorm]
  swap; · rw [← memℒp_one_iff_integrable]; exact Lp.memℒp _
  have h_eq :
    ∫⁻ a, ‖condexpIndL1Fin hm hs hμs x a‖₊ ∂μ = ∫⁻ a, ‖condexpIndSMul hm hs hμs x a‖₊ ∂μ := by
    

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
this : 0 ≤ ∫ (a : α), ‖↑↑(condexpIndL1Fin hm hs hμs x) a‖ ∂μ
⊢ ∫⁻ (a : α), ↑‖↑↑(condexpIndL1Fin hm hs hμs x) a‖₊ ∂μ = ∫⁻ (a : α), ↑‖↑↑(condexpIndSMul hm hs hμs x) a‖₊ ∂μ

Next tactic:
refine' lintegral_congr_ae _
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
  have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ :=
    integral_nonneg fun a => norm_nonneg _
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
    ENNReal.toReal_ofReal this,
    ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_nnnorm]
  swap; · rw [← memℒp_one_iff_integrable]; exact Lp.memℒp _
  have h_eq :
    ∫⁻ a, ‖condexpIndL1Fin hm hs hμs x a‖₊ ∂μ = ∫⁻ a, ‖condexpIndSMul hm hs hμs x a‖₊ ∂μ := by
    refine' lintegral_congr_ae _
    

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
this : 0 ≤ ∫ (a : α), ‖↑↑(condexpIndL1Fin hm hs hμs x) a‖ ∂μ
⊢ (fun a => ↑‖↑↑(condexpIndL1Fin hm hs hμs x) a‖₊) =ᵐ[μ] fun a => ↑‖↑↑(condexpIndSMul hm hs hμs x) a‖₊

Next tactic:
refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun z hz => _
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
  have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ :=
    integral_nonneg fun a => norm_nonneg _
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
    ENNReal.toReal_ofReal this,
    ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_nnnorm]
  swap; · rw [← memℒp_one_iff_integrable]; exact Lp.memℒp _
  have h_eq :
    ∫⁻ a, ‖condexpIndL1Fin hm hs hμs x a‖₊ ∂μ = ∫⁻ a, ‖condexpIndSMul hm hs hμs x a‖₊ ∂μ := by
    refine' lintegral_congr_ae _
    refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun z hz => _
    

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
this : 0 ≤ ∫ (a : α), ‖↑↑(condexpIndL1Fin hm hs hμs x) a‖ ∂μ
z : α
hz : ↑↑(condexpIndL1Fin hm hs hμs x) z = ↑↑(condexpIndSMul hm hs hμs x) z
⊢ (fun a => ↑‖↑↑(condexpIndL1Fin hm hs hμs x) a‖₊) z = (fun a => ↑‖↑↑(condexpIndSMul hm hs hμs x) a‖₊) z

Next tactic:
dsimp only
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
  have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ :=
    integral_nonneg fun a => norm_nonneg _
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
    ENNReal.toReal_ofReal this,
    ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_nnnorm]
  swap; · rw [← memℒp_one_iff_integrable]; exact Lp.memℒp _
  have h_eq :
    ∫⁻ a, ‖condexpIndL1Fin hm hs hμs x a‖₊ ∂μ = ∫⁻ a, ‖condexpIndSMul hm hs hμs x a‖₊ ∂μ := by
    refine' lintegral_congr_ae _
    refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun z hz => _
    dsimp only
    

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
this : 0 ≤ ∫ (a : α), ‖↑↑(condexpIndL1Fin hm hs hμs x) a‖ ∂μ
z : α
hz : ↑↑(condexpIndL1Fin hm hs hμs x) z = ↑↑(condexpIndSMul hm hs hμs x) z
⊢ ↑‖↑↑(condexpIndL1Fin hm hs hμs x) z‖₊ = ↑‖↑↑(condexpIndSMul hm hs hμs x) z‖₊

Next tactic:
rw [hz]
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
  have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ :=
    integral_nonneg fun a => norm_nonneg _
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
    ENNReal.toReal_ofReal this,
    ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_nnnorm]
  swap; · rw [← memℒp_one_iff_integrable]; exact Lp.memℒp _
  have h_eq :
    ∫⁻ a, ‖condexpIndL1Fin hm hs hμs x a‖₊ ∂μ = ∫⁻ a, ‖condexpIndSMul hm hs hμs x a‖₊ ∂μ := by
    refine' lintegral_congr_ae _
    refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun z hz => _
    dsimp only
    rw [hz]
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
this : 0 ≤ ∫ (a : α), ‖↑↑(condexpIndL1Fin hm hs hμs x) a‖ ∂μ
h_eq : ∫⁻ (a : α), ↑‖↑↑(condexpIndL1Fin hm hs hμs x) a‖₊ ∂μ = ∫⁻ (a : α), ↑‖↑↑(condexpIndSMul hm hs hμs x) a‖₊ ∂μ
⊢ ∫⁻ (x_1 : α), ↑‖↑↑(condexpIndL1Fin hm hs hμs x) x_1‖₊ ∂μ ≤ ↑↑μ s * ENNReal.ofReal ‖x‖

Next tactic:
rw [h_eq, ofReal_norm_eq_coe_nnnorm]
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
  have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ :=
    integral_nonneg fun a => norm_nonneg _
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
    ENNReal.toReal_ofReal this,
    ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_nnnorm]
  swap; · rw [← memℒp_one_iff_integrable]; exact Lp.memℒp _
  have h_eq :
    ∫⁻ a, ‖condexpIndL1Fin hm hs hμs x a‖₊ ∂μ = ∫⁻ a, ‖condexpIndSMul hm hs hμs x a‖₊ ∂μ := by
    refine' lintegral_congr_ae _
    refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun z hz => _
    dsimp only
    rw [hz]
  rw [h_eq, ofReal_norm_eq_coe_nnnorm]
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
this : 0 ≤ ∫ (a : α), ‖↑↑(condexpIndL1Fin hm hs hμs x) a‖ ∂μ
h_eq : ∫⁻ (a : α), ↑‖↑↑(condexpIndL1Fin hm hs hμs x) a‖₊ ∂μ = ∫⁻ (a : α), ↑‖↑↑(condexpIndSMul hm hs hμs x) a‖₊ ∂μ
⊢ ∫⁻ (a : α), ↑‖↑↑(condexpIndSMul hm hs hμs x) a‖₊ ∂μ ≤ ↑↑μ s * ↑‖x‖₊

Next tactic:
exact lintegral_nnnorm_condexpIndSMul_le hm hs hμs x
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
⊢ condexpIndL1Fin hm (_ : MeasurableSet (s ∪ t)) (_ : ↑↑μ (s ∪ t) ≠ ⊤) x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x

Next tactic:
ext1
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
⊢ ↑↑(condexpIndL1Fin hm (_ : MeasurableSet (s ∪ t)) (_ : ↑↑μ (s ∪ t) ≠ ⊤) x) =ᵐ[μ]
    ↑↑(condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x)

Next tactic:
have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
⊢ ↑↑(condexpIndL1Fin hm (_ : MeasurableSet (s ∪ t)) (_ : ↑↑μ (s ∪ t) ≠ ⊤) x) =ᵐ[μ]
    ↑↑(condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x)

Next tactic:
refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
⊢ ↑↑(condexpIndSMul hm (_ : MeasurableSet (s ∪ t)) hμst x) =ᵐ[μ]
    ↑↑(condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x)

Next tactic:
refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
⊢ ↑↑(condexpIndSMul hm (_ : MeasurableSet (s ∪ t)) hμst x) =ᵐ[μ]
    ↑↑(condexpIndL1Fin hm hs hμs x) + ↑↑(condexpIndL1Fin hm ht hμt x)

Next tactic:
have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
⊢ ↑↑(condexpIndSMul hm (_ : MeasurableSet (s ∪ t)) hμst x) =ᵐ[μ]
    ↑↑(condexpIndL1Fin hm hs hμs x) + ↑↑(condexpIndL1Fin hm ht hμt x)

Next tactic:
have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
ht_eq : ↑↑(condexpIndL1Fin hm ht hμt x) =ᵐ[μ] ↑↑(condexpIndSMul hm ht hμt x)
⊢ ↑↑(condexpIndSMul hm (_ : MeasurableSet (s ∪ t)) hμst x) =ᵐ[μ]
    ↑↑(condexpIndL1Fin hm hs hμs x) + ↑↑(condexpIndL1Fin hm ht hμt x)

Next tactic:
refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)