Theorem: ContinuousMap.toLp_denseRange
File: Mathlib/MeasureTheory/Function/ContinuousMapDense.lean

tactic: haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
state_before: α : Type u_1
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : NormalSpace α
inst✝⁸ : BorelSpace α
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
μ : MeasureTheory.Measure α
p : ℝ≥0∞
inst✝⁶ : SecondCountableTopologyEither α E
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
𝕜 : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedAlgebra ℝ 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompactSpace α
inst✝¹ : Measure.WeaklyRegular μ
inst✝ : IsFiniteMeasure μ
⊢ DenseRange ↑(toLp p μ 𝕜)

tactic: rw [denseRange_iff_closure_range]
state_before: α : Type u_1
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : NormalSpace α
inst✝⁸ : BorelSpace α
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
μ : MeasureTheory.Measure α
p : ℝ≥0∞
inst✝⁶ : SecondCountableTopologyEither α E
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
𝕜 : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedAlgebra ℝ 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompactSpace α
inst✝¹ : Measure.WeaklyRegular μ
inst✝ : IsFiniteMeasure μ
this : NormedSpace ℝ E
⊢ DenseRange ↑(toLp p μ 𝕜)

tactic: suffices (LinearMap.range (toLp p μ 𝕜 : _ →L[𝕜] Lp E p μ)).toAddSubgroup.topologicalClosure = ⊤
  by exact congr_arg ((↑) : AddSubgroup (Lp E p μ) → Set (Lp E p μ)) this
state_before: α : Type u_1
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : NormalSpace α
inst✝⁸ : BorelSpace α
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
μ : MeasureTheory.Measure α
p : ℝ≥0∞
inst✝⁶ : SecondCountableTopologyEither α E
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
𝕜 : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedAlgebra ℝ 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompactSpace α
inst✝¹ : Measure.WeaklyRegular μ
inst✝ : IsFiniteMeasure μ
this : NormedSpace ℝ E
⊢ closure (range ↑(toLp p μ 𝕜)) = univ

tactic: simpa [range_toLp p μ] using MeasureTheory.Lp.boundedContinuousFunction_dense E hp
state_before: α : Type u_1
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : NormalSpace α
inst✝⁸ : BorelSpace α
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
μ : MeasureTheory.Measure α
p : ℝ≥0∞
inst✝⁶ : SecondCountableTopologyEither α E
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
𝕜 : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedAlgebra ℝ 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompactSpace α
inst✝¹ : Measure.WeaklyRegular μ
inst✝ : IsFiniteMeasure μ
this : NormedSpace ℝ E
⊢ AddSubgroup.topologicalClosure (Submodule.toAddSubgroup (LinearMap.range (toLp p μ 𝕜))) = ⊤

tactic: exact congr_arg ((↑) : AddSubgroup (Lp E p μ) → Set (Lp E p μ)) this
state_before: α : Type u_1
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : NormalSpace α
inst✝⁸ : BorelSpace α
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
μ : MeasureTheory.Measure α
p : ℝ≥0∞
inst✝⁶ : SecondCountableTopologyEither α E
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
𝕜 : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedAlgebra ℝ 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompactSpace α
inst✝¹ : Measure.WeaklyRegular μ
inst✝ : IsFiniteMeasure μ
this✝ : NormedSpace ℝ E
this : AddSubgroup.topologicalClosure (Submodule.toAddSubgroup (LinearMap.range (toLp p μ 𝕜))) = ⊤
⊢ closure (range ↑(toLp p μ 𝕜)) = univ