l3lab/ntp-mathlib huggingface dataset:
theorem setToL1_congr_left' (T T' : Set α → E →L[ℝ] F) {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁[μ] E) :
    setToL1 hT f = setToL1 hT' f := by
  suffices setToL1 hT = setToL1 hT' by rw [this]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ _
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T✝ T'✝ T'' : Set α → E →L[ℝ] F
C✝ C'✝ C'' : ℝ
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = T' s
f : ↥(Lp E 1)
⊢ ContinuousLinearMap.comp (setToL1 hT') (coeToLp α E ℝ) = setToL1SCLM α E μ hT

Next tactic:
ext1 f
theorem setToL1_congr_left' (T T' : Set α → E →L[ℝ] F) {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁[μ] E) :
    setToL1 hT f = setToL1 hT' f := by
  suffices setToL1 hT = setToL1 hT' by rw [this]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ _
  ext1 f
  

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T✝ T'✝ T'' : Set α → E →L[ℝ] F
C✝ C'✝ C'' : ℝ
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = T' s
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
⊢ (ContinuousLinearMap.comp (setToL1 hT') (coeToLp α E ℝ)) f = (setToL1SCLM α E μ hT) f

Next tactic:
suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]; rfl
theorem setToL1_congr_left' (T T' : Set α → E →L[ℝ] F) {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁[μ] E) :
    setToL1 hT f = setToL1 hT' f := by
  suffices setToL1 hT = setToL1 hT' by rw [this]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ _
  ext1 f
  suffices setToL1 hT' f = setToL1SCLM α E μ hT f by 

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T✝ T'✝ T'' : Set α → E →L[ℝ] F
C✝ C'✝ C'' : ℝ
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = T' s
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
this : (setToL1 hT') ↑f = (setToL1SCLM α E μ hT) f
⊢ (ContinuousLinearMap.comp (setToL1 hT') (coeToLp α E ℝ)) f = (setToL1SCLM α E μ hT) f

Next tactic:
rw [← this]
theorem setToL1_congr_left' (T T' : Set α → E →L[ℝ] F) {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁[μ] E) :
    setToL1 hT f = setToL1 hT' f := by
  suffices setToL1 hT = setToL1 hT' by rw [this]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ _
  ext1 f
  suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]; 

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T✝ T'✝ T'' : Set α → E →L[ℝ] F
C✝ C'✝ C'' : ℝ
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = T' s
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
this : (setToL1 hT') ↑f = (setToL1SCLM α E μ hT) f
⊢ (ContinuousLinearMap.comp (setToL1 hT') (coeToLp α E ℝ)) f = (setToL1 hT') ↑f

Next tactic:
rfl
theorem setToL1_congr_left' (T T' : Set α → E →L[ℝ] F) {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁[μ] E) :
    setToL1 hT f = setToL1 hT' f := by
  suffices setToL1 hT = setToL1 hT' by rw [this]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ _
  ext1 f
  suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]; rfl
  

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T✝ T'✝ T'' : Set α → E →L[ℝ] F
C✝ C'✝ C'' : ℝ
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = T' s
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
⊢ (setToL1 hT') ↑f = (setToL1SCLM α E μ hT) f

Next tactic:
rw [setToL1_eq_setToL1SCLM]
theorem setToL1_congr_left' (T T' : Set α → E →L[ℝ] F) {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁[μ] E) :
    setToL1 hT f = setToL1 hT' f := by
  suffices setToL1 hT = setToL1 hT' by rw [this]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ _
  ext1 f
  suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]; rfl
  rw [setToL1_eq_setToL1SCLM]
  

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T✝ T'✝ T'' : Set α → E →L[ℝ] F
C✝ C'✝ C'' : ℝ
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = T' s
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
⊢ (setToL1SCLM α E μ hT') f = (setToL1SCLM α E μ hT) f

Next tactic:
exact (setToL1SCLM_congr_left' hT hT' h f).symm
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
    setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
f : ↥(Lp E 1)
⊢ (setToL1 (_ : DominatedFinMeasAdditive μ (T + T') (C + C'))) f = (setToL1 hT) f + (setToL1 hT') f

Next tactic:
suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by
    rw [this, ContinuousLinearMap.add_apply]
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
    setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by
    

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
f : ↥(Lp E 1)
this : setToL1 (_ : DominatedFinMeasAdditive μ (T + T') (C + C')) = setToL1 hT + setToL1 hT'
⊢ (setToL1 (_ : DominatedFinMeasAdditive μ (T + T') (C + C'))) f = (setToL1 hT) f + (setToL1 hT') f

Next tactic:
rw [this, ContinuousLinearMap.add_apply]
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
    setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by
    rw [this, ContinuousLinearMap.add_apply]
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
f : ↥(Lp E 1)
⊢ setToL1 (_ : DominatedFinMeasAdditive μ (T + T') (C + C')) = setToL1 hT + setToL1 hT'

Next tactic:
refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.add hT')) _ _ _ _ _
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
    setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by
    rw [this, ContinuousLinearMap.add_apply]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.add hT')) _ _ _ _ _
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
f : ↥(Lp E 1)
⊢ ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp α E ℝ) =
    setToL1SCLM α E μ (_ : DominatedFinMeasAdditive μ (T + T') (C + C'))

Next tactic:
ext1 f
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
    setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by
    rw [this, ContinuousLinearMap.add_apply]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.add hT')) _ _ _ _ _
  ext1 f
  

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
⊢ (ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp α E ℝ)) f =
    (setToL1SCLM α E μ (_ : DominatedFinMeasAdditive μ (T + T') (C + C'))) f

Next tactic:
suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM α E μ (hT.add hT') f by
    rw [← this]; rfl
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
    setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by
    rw [this, ContinuousLinearMap.add_apply]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.add hT')) _ _ _ _ _
  ext1 f
  suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM α E μ (hT.add hT') f by
    

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
this : (setToL1 hT) ↑f + (setToL1 hT') ↑f = (setToL1SCLM α E μ (_ : DominatedFinMeasAdditive μ (T + T') (C + C'))) f
⊢ (ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp α E ℝ)) f =
    (setToL1SCLM α E μ (_ : DominatedFinMeasAdditive μ (T + T') (C + C'))) f

Next tactic:
rw [← this]
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
    setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by
    rw [this, ContinuousLinearMap.add_apply]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.add hT')) _ _ _ _ _
  ext1 f
  suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM α E μ (hT.add hT') f by
    rw [← this]; 

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
this : (setToL1 hT) ↑f + (setToL1 hT') ↑f = (setToL1SCLM α E μ (_ : DominatedFinMeasAdditive μ (T + T') (C + C'))) f
⊢ (ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp α E ℝ)) f = (setToL1 hT) ↑f + (setToL1 hT') ↑f

Next tactic:
rfl
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
    setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by
    rw [this, ContinuousLinearMap.add_apply]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.add hT')) _ _ _ _ _
  ext1 f
  suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM α E μ (hT.add hT') f by
    rw [← this]; rfl
  

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
⊢ (setToL1 hT) ↑f + (setToL1 hT') ↑f = (setToL1SCLM α E μ (_ : DominatedFinMeasAdditive μ (T + T') (C + C'))) f

Next tactic:
rw [setToL1_eq_setToL1SCLM, setToL1_eq_setToL1SCLM, setToL1SCLM_add_left hT hT']
theorem setToL1_add_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
    (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁[μ] E) :
    setToL1 hT'' f = setToL1 hT f + setToL1 hT' f := by
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hT'' : DominatedFinMeasAdditive μ T'' C''
h_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s
f : ↥(Lp E 1)
⊢ (setToL1 hT'') f = (setToL1 hT) f + (setToL1 hT') f

Next tactic:
suffices setToL1 hT'' = setToL1 hT + setToL1 hT' by rw [this, ContinuousLinearMap.add_apply]
theorem setToL1_add_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
    (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁[μ] E) :
    setToL1 hT'' f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 hT'' = setToL1 hT + setToL1 hT' by 

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hT'' : DominatedFinMeasAdditive μ T'' C''
h_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s
f : ↥(Lp E 1)
this : setToL1 hT'' = setToL1 hT + setToL1 hT'
⊢ (setToL1 hT'') f = (setToL1 hT) f + (setToL1 hT') f

Next tactic:
rw [this, ContinuousLinearMap.add_apply]
theorem setToL1_add_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
    (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁[μ] E) :
    setToL1 hT'' f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 hT'' = setToL1 hT + setToL1 hT' by rw [this, ContinuousLinearMap.add_apply]
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hT'' : DominatedFinMeasAdditive μ T'' C''
h_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s
f : ↥(Lp E 1)
⊢ setToL1 hT'' = setToL1 hT + setToL1 hT'

Next tactic:
refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT'') _ _ _ _ _
theorem setToL1_add_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
    (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁[μ] E) :
    setToL1 hT'' f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 hT'' = setToL1 hT + setToL1 hT' by rw [this, ContinuousLinearMap.add_apply]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT'') _ _ _ _ _
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hT'' : DominatedFinMeasAdditive μ T'' C''
h_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s
f : ↥(Lp E 1)
⊢ ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp α E ℝ) = setToL1SCLM α E μ hT''

Next tactic:
ext1 f
theorem setToL1_add_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
    (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁[μ] E) :
    setToL1 hT'' f = setToL1 hT f + setToL1 hT' f := by
  suffices setToL1 hT'' = setToL1 hT + setToL1 hT' by rw [this, ContinuousLinearMap.add_apply]
  refine' ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT'') _ _ _ _ _
  ext1 f
  

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hT'' : DominatedFinMeasAdditive μ T'' C''
h_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s
f✝ : ↥(Lp E 1)
f : ↥(simpleFunc E 1 μ)
⊢ (ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp α E ℝ)) f = (setToL1SCLM α E μ hT'') f

Next tactic:
suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM α E μ hT'' f by rw [← this]; congr