Theorem: ContinuousMap.range_toLp
File: Mathlib/MeasureTheory/Function/LpSpace.lean

tactic: refine' SetLike.ext' _
state_before: α : Type u_3
E : Type u_2
F : Type ?u.13639271
G : Type ?u.13639274
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : MeasureTheory.Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : TopologicalSpace α
inst✝⁶ : BorelSpace α
inst✝⁵ : SecondCountableTopologyEither α E
inst✝⁴ : CompactSpace α
inst✝³ : IsFiniteMeasure μ
𝕜 : Type u_1
inst✝² : Fact (1 ≤ p)
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
⊢ Submodule.toAddSubgroup (LinearMap.range (toLp p μ 𝕜)) = Lp.boundedContinuousFunction E p μ

tactic: have := (linearIsometryBoundedOfCompact α E 𝕜).surjective
state_before: α : Type u_3
E : Type u_2
F : Type ?u.13639271
G : Type ?u.13639274
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : MeasureTheory.Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : TopologicalSpace α
inst✝⁶ : BorelSpace α
inst✝⁵ : SecondCountableTopologyEither α E
inst✝⁴ : CompactSpace α
inst✝³ : IsFiniteMeasure μ
𝕜 : Type u_1
inst✝² : Fact (1 ≤ p)
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
⊢ ↑(Submodule.toAddSubgroup (LinearMap.range (toLp p μ 𝕜))) = ↑(Lp.boundedContinuousFunction E p μ)

tactic: convert Function.Surjective.range_comp this (BoundedContinuousFunction.toLp (E := E) p μ 𝕜)
state_before: α : Type u_3
E : Type u_2
F : Type ?u.13639271
G : Type ?u.13639274
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : MeasureTheory.Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : TopologicalSpace α
inst✝⁶ : BorelSpace α
inst✝⁵ : SecondCountableTopologyEither α E
inst✝⁴ : CompactSpace α
inst✝³ : IsFiniteMeasure μ
𝕜 : Type u_1
inst✝² : Fact (1 ≤ p)
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
this : Function.Surjective ↑(linearIsometryBoundedOfCompact α E 𝕜)
⊢ ↑(Submodule.toAddSubgroup (LinearMap.range (toLp p μ 𝕜))) = ↑(Lp.boundedContinuousFunction E p μ)

tactic: rw [← BoundedContinuousFunction.range_toLp p μ (𝕜 := 𝕜)]
state_before: case h.e'_3
α : Type u_3
E : Type u_2
F : Type ?u.13639271
G : Type ?u.13639274
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : MeasureTheory.Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : TopologicalSpace α
inst✝⁶ : BorelSpace α
inst✝⁵ : SecondCountableTopologyEither α E
inst✝⁴ : CompactSpace α
inst✝³ : IsFiniteMeasure μ
𝕜 : Type u_1
inst✝² : Fact (1 ≤ p)
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
this : Function.Surjective ↑(linearIsometryBoundedOfCompact α E 𝕜)
⊢ ↑(Lp.boundedContinuousFunction E p μ) = Set.range ↑(BoundedContinuousFunction.toLp p μ 𝕜)

tactic: rfl
state_before: case h.e'_3
α : Type u_3
E : Type u_2
F : Type ?u.13639271
G : Type ?u.13639274
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : MeasureTheory.Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : TopologicalSpace α
inst✝⁶ : BorelSpace α
inst✝⁵ : SecondCountableTopologyEither α E
inst✝⁴ : CompactSpace α
inst✝³ : IsFiniteMeasure μ
𝕜 : Type u_1
inst✝² : Fact (1 ≤ p)
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
this : Function.Surjective ↑(linearIsometryBoundedOfCompact α E 𝕜)
⊢ ↑(Submodule.toAddSubgroup (LinearMap.range (BoundedContinuousFunction.toLp p μ 𝕜))) =
    Set.range ↑(BoundedContinuousFunction.toLp p μ 𝕜)