Theorem: MeasureTheory.SignedMeasure.singularPart_neg
File: Mathlib/MeasureTheory/Decomposition/SignedLebesgue.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_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ (-s).singularPart μ = -s.singularPart μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  ((-s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
⊢ (-s).singularPart μ = -s.singularPart 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_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  ((-s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
⊢ (-s).singularPart μ = -s.singularPart μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  ((-s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
h₂ :
  ((-s).toJordanDecomposition.negPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure
⊢ (-s).singularPart μ = -s.singularPart μ

tactic: rw [singularPart, singularPart, neg_sub, h₁, h₂]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  ((-s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
h₂ :
  ((-s).toJordanDecomposition.negPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure
⊢ (-s).singularPart μ = -s.singularPart μ
state_after: no goals

tactic: refine toSignedMeasure_congr ?_
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ ((-s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ (-s).toJordanDecomposition.posPart.singularPart μ = s.toJordanDecomposition.negPart.singularPart μ

tactic: rw [toJordanDecomposition_neg, JordanDecomposition.neg_posPart]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
⊢ (-s).toJordanDecomposition.posPart.singularPart μ = s.toJordanDecomposition.negPart.singularPart μ
state_after: no goals

tactic: refine toSignedMeasure_congr ?_
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  ((-s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
⊢ ((-s).toJordanDecomposition.negPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  ((-s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
⊢ (-s).toJordanDecomposition.negPart.singularPart μ = s.toJordanDecomposition.posPart.singularPart μ

tactic: rw [toJordanDecomposition_neg, JordanDecomposition.neg_negPart]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
h₁ :
  ((-s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure =
    (s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure
⊢ (-s).toJordanDecomposition.negPart.singularPart μ = s.toJordanDecomposition.posPart.singularPart μ
state_after: no goals