-- Mathlib/MeasureTheory/Function/LpSpace/ContinuousCompMeasurePreserving.lean
theorem compMeasurePreserving_continuous (hp : p ≠ ∞) :
    Continuous fun gf : Lp E p ν × {f : C(X, Y) // MeasurePreserving f μ ν} ↦
      compMeasurePreserving gf.2.1 gf.2.2 gf.1

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_3
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ⊤
⊢ Continuous fun gf => (compMeasurePreserving ⇑↑gf.2 ⋯) gf.1

Tactic:
have hp₀ : p ≠ 0 := (one_pos.trans_le Fact.out).ne'

Result:
X : Type u_1
Y : Type u_2
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_3
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ⊤
hp₀ : p ≠ 0
⊢ Continuous fun gf => (compMeasurePreserving ⇑↑gf.2 ⋯) gf.1

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_3
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ⊤
hp₀ : p ≠ 0
f : ↥(Lp E p ν)
hf : f ∈ ↑(simpleFunc E p ν)
⊢ Continuous fun y => (compMeasurePreserving ⇑↑(f, y).2 ⋯) (f, y).1

Tactic:
lift f to Lp.simpleFunc E p ν using hf

Result:
case intro
X : Type u_1
Y : Type u_2
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_3
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ⊤
hp₀ : p ≠ 0
f : ↥(simpleFunc E p ν)
⊢ Continuous fun y => (compMeasurePreserving ⇑↑(↑f, y).2 ⋯) (↑f, y).1

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_3
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ⊤
hp₀ : p ≠ 0
⊢ ∀ a ∈ ↑(simpleFunc E p ν), Continuous fun y => (compMeasurePreserving ⇑↑(a, y).2 ⋯) (a, y).1

Tactic:
intro f hf

Result:
X : Type u_1
Y : Type u_2
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_3
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ⊤
hp₀ : p ≠ 0
f : ↥(Lp E p ν)
hf : f ∈ ↑(simpleFunc E p ν)
⊢ Continuous fun y => (compMeasurePreserving ⇑↑(f, y).2 ⋯) (f, y).1