-- Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
theorem condExpL1CLM_lpMeas (f : lpMeas F' ℝ m 1 μ) :
    condExpL1CLM F' hm μ (f : α →₁[μ] F') = ↑f

Proof state:
case refine_1
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g
c : F'
s : Set α
hs : MeasurableSet s
hμs : (μ.trim hm) s < ⊤
⊢ (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm ↑(simpleFunc.indicatorConst 1 hs ⋯ c)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm ↑(simpleFunc.indicatorConst 1 hs ⋯ c))

Tactic:
rw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator hs hμs.ne c,
  condExpL1CLM_indicatorConstLp]

Result:
case refine_1
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g
c : F'
s : Set α
hs : MeasurableSet s
hμs : (μ.trim hm) s < ⊤
⊢ (condExpInd F' hm μ s) c = indicatorConstLp 1 ⋯ ⋯ c

Proof state:
case refine_3.refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g
⊢ Continuous fun f => ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm f)

Tactic:
refine continuous_induced_dom.comp ?_

Result:
case refine_3.refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g
⊢ Continuous ⇑(lpMeasToLpTrimLie F' ℝ 1 μ hm).symm

Proof state:
case refine_3.refine_1
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g
⊢ Continuous ⇑(lpMeasToLpTrimLie F' ℝ 1 μ hm).symm

Tactic:
exact LinearIsometryEquiv.continuous _

Result:
no goals

Proof state:
case refine_3.refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g
⊢ Continuous ⇑(lpMeasToLpTrimLie F' ℝ 1 μ hm).symm

Tactic:
exact LinearIsometryEquiv.continuous _

Result:
no goals

Proof state:
case refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g
⊢ ∀ ⦃f g : α → F'⦄ (hf : MemLp f 1 (μ.trim hm)) (hg : MemLp g 1 (μ.trim hm)),
    Disjoint (Function.support f) (Function.support g) →
      (fun g =>
            (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g) =
              ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g))
          (MemLp.toLp f hf) →
        (fun g =>
              (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g) =
                ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g))
            (MemLp.toLp g hg) →
          (fun g =>
              (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g) =
                ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g))
            (MemLp.toLp f hf + MemLp.toLp g hg)

Tactic:
intro f g hf hg _ hf_eq hg_eq

Result:
case refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f✝ : ↥(lpMeas F' ℝ m 1 μ)
g✝ : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f✝
hfg : f✝ = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g✝
f g : α → F'
hf : MemLp f 1 (μ.trim hm)
hg : MemLp g 1 (μ.trim hm)
a✝ : Disjoint (Function.support f) (Function.support g)
hf_eq :
  (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf))
hg_eq :
  (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg))
⊢ (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg))

Proof state:
case refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f✝ : ↥(lpMeas F' ℝ m 1 μ)
g✝ : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f✝
hfg : f✝ = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g✝
f g : α → F'
hf : MemLp f 1 (μ.trim hm)
hg : MemLp g 1 (μ.trim hm)
a✝ : Disjoint (Function.support f) (Function.support g)
hf_eq :
  (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf))
hg_eq :
  (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg))
⊢ (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf + MemLp.toLp g hg))

Tactic:
rw [LinearIsometryEquiv.map_add]

Result:
case refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f✝ : ↥(lpMeas F' ℝ m 1 μ)
g✝ : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f✝
hfg : f✝ = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g✝
f g : α → F'
hf : MemLp f 1 (μ.trim hm)
hg : MemLp g 1 (μ.trim hm)
a✝ : Disjoint (Function.support f) (Function.support g)
hf_eq :
  (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf))
hg_eq :
  (condExpL1CLM F' hm μ) ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg))
⊢ (condExpL1CLM F' hm μ)
      ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf) +
          (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg)) =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp f hf) + (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm (MemLp.toLp g hg))