Theorem: MeasureTheory.ComplexMeasure.equivSignedMeasureₗ_symm_apply
File: Mathlib/MeasureTheory/Measure/Complex.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
R : Type u_3
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ : ContinuousConstSMul R ℝ
inst✝ : ContinuousConstSMul R ℂ
c d : ComplexMeasure α
⊢ __src✝.toFun (c + d) = __src✝.toFun c + __src✝.toFun d
state_after: no goals
tactic: rfl
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
R : Type u_3
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ : ContinuousConstSMul R ℝ
inst✝ : ContinuousConstSMul R ℂ
⊢ ∀ (m_1 : R) (x : ComplexMeasure α),
    { toFun := __src✝.toFun, map_add' := ⋯ }.toFun (m_1 • x) =
      (RingHom.id R) m_1 • { toFun := __src✝.toFun, map_add' := ⋯ }.toFun x
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
R : Type u_3
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ : ContinuousConstSMul R ℝ
inst✝ : ContinuousConstSMul R ℂ
r : R
c : ComplexMeasure α
⊢ { toFun := __src✝.toFun, map_add' := ⋯ }.toFun (r • c) =
    (RingHom.id R) r • { toFun := __src✝.toFun, map_add' := ⋯ }.toFun c
tactic: intro r c
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
R : Type u_3
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ : ContinuousConstSMul R ℝ
inst✝ : ContinuousConstSMul R ℂ
r : R
c : ComplexMeasure α
⊢ { toFun := __src✝.toFun, map_add' := ⋯ }.toFun (r • c) =
    (RingHom.id R) r • { toFun := __src✝.toFun, map_add' := ⋯ }.toFun c
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
R : Type u_3
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ : ContinuousConstSMul R ℝ
inst✝ : ContinuousConstSMul R ℂ
r : R
c : ComplexMeasure α
⊢ (mapRange (r • c) Complex.reLm.toAddMonoidHom Complex.continuous_re,
      mapRange (r • c) Complex.imLm.toAddMonoidHom Complex.continuous_im) =
    (r • mapRange c Complex.reLm.toAddMonoidHom Complex.continuous_re,
      r • mapRange c Complex.imLm.toAddMonoidHom Complex.continuous_im)
tactic: dsimp
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
R : Type u_3
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ : ContinuousConstSMul R ℝ
inst✝ : ContinuousConstSMul R ℂ
r : R
c : ComplexMeasure α
⊢ (mapRange (r • c) Complex.reLm.toAddMonoidHom Complex.continuous_re,
      mapRange (r • c) Complex.imLm.toAddMonoidHom Complex.continuous_im) =
    (r • mapRange c Complex.reLm.toAddMonoidHom Complex.continuous_re,
      r • mapRange c Complex.imLm.toAddMonoidHom Complex.continuous_im)
state_after: case a.h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
R : Type u_3
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ : ContinuousConstSMul R ℝ
inst✝ : ContinuousConstSMul R ℂ
r : R
c : ComplexMeasure α
i✝ : Set α
a✝ : MeasurableSet i✝
⊢ ↑(mapRange (r • c) Complex.reLm.toAddMonoidHom Complex.continuous_re,
            mapRange (r • c) Complex.imLm.toAddMonoidHom Complex.continuous_im).1
      i✝ =
    ↑(r • mapRange c Complex.reLm.toAddMonoidHom Complex.continuous_re,
            r • mapRange c Complex.imLm.toAddMonoidHom Complex.continuous_im).1
      i✝

case a.h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
R : Type u_3
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ : ContinuousConstSMul R ℝ
inst✝ : ContinuousConstSMul R ℂ
r : R
c : ComplexMeasure α
i✝ : Set α
a✝ : MeasurableSet i✝
⊢ ↑(mapRange (r • c) Complex.reLm.toAddMonoidHom Complex.continuous_re,
            mapRange (r • c) Complex.imLm.toAddMonoidHom Complex.continuous_im).2
      i✝ =
    ↑(r • mapRange c Complex.reLm.toAddMonoidHom Complex.continuous_re,
            r • mapRange c Complex.imLm.toAddMonoidHom Complex.continuous_im).2
      i✝
tactic: ext