Theorem: MeasureTheory.SignedMeasure.singularPart_neg
File: Mathlib/MeasureTheory/Decomposition/Lebesgue.lean

tactic: have h₁ :
  ((-s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure := by
  refine' toSignedMeasure_congr _
  rw [toJordanDecomposition_neg, JordanDecomposition.neg_posPart]
state_before: α : Type u_1
β : Type ?u.622814
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ singularPart (-s) μ = -singularPart s μ

tactic: have h₂ :
  ((-s).toJordanDecomposition.negPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure := by
  refine' toSignedMeasure_congr _
  rw [toJordanDecomposition_neg, JordanDecomposition.neg_negPart]
state_before: α : Type u_1
β : Type ?u.622814
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  toSignedMeasure (Measure.singularPart (toJordanDecomposition (-s)).posPart μ) =
    toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart μ)
⊢ singularPart (-s) μ = -singularPart s μ

tactic: rw [singularPart, singularPart, neg_sub, h₁, h₂]
state_before: α : Type u_1
β : Type ?u.622814
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  toSignedMeasure (Measure.singularPart (toJordanDecomposition (-s)).posPart μ) =
    toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart μ)
h₂ :
  toSignedMeasure (Measure.singularPart (toJordanDecomposition (-s)).negPart μ) =
    toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart μ)
⊢ singularPart (-s) μ = -singularPart s μ

tactic: refine' toSignedMeasure_congr _
state_before: α : Type u_1
β : Type ?u.622814
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ toSignedMeasure (Measure.singularPart (toJordanDecomposition (-s)).posPart μ) =
    toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart μ)

tactic: rw [toJordanDecomposition_neg, JordanDecomposition.neg_posPart]
state_before: α : Type u_1
β : Type ?u.622814
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ Measure.singularPart (toJordanDecomposition (-s)).posPart μ = Measure.singularPart (toJordanDecomposition s).negPart μ

tactic: refine' toSignedMeasure_congr _
state_before: α : Type u_1
β : Type ?u.622814
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  toSignedMeasure (Measure.singularPart (toJordanDecomposition (-s)).posPart μ) =
    toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart μ)
⊢ toSignedMeasure (Measure.singularPart (toJordanDecomposition (-s)).negPart μ) =
    toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart μ)

tactic: rw [toJordanDecomposition_neg, JordanDecomposition.neg_negPart]
state_before: α : Type u_1
β : Type ?u.622814
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  toSignedMeasure (Measure.singularPart (toJordanDecomposition (-s)).posPart μ) =
    toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart μ)
⊢ Measure.singularPart (toJordanDecomposition (-s)).negPart μ = Measure.singularPart (toJordanDecomposition s).posPart μ