Theorem: MeasureTheory.Measure.QuasiMeasurePreserving.image_zpow_ae_eq
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: rw [Equiv.image_eq_preimage]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℤ
hs : ⇑e '' s =ᵐ[μ] s
⊢ ⇑(e ^ k) '' s =ᵐ[μ] s
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℤ
hs : ⇑e '' s =ᵐ[μ] s
⊢ ⇑(e ^ k).symm ⁻¹' s =ᵐ[μ] s

tactic: obtain ⟨k, rfl | rfl⟩ := k.eq_nat_or_neg
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℤ
hs : ⇑e '' s =ᵐ[μ] s
⊢ ⇑(e ^ k).symm ⁻¹' s =ᵐ[μ] s
state_after: case intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ ↑k).symm ⁻¹' s =ᵐ[μ] s

case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ (-↑k)).symm ⁻¹' s =ᵐ[μ] s

tactic: replace hs : (⇑e⁻¹) ⁻¹' s =ᵐ[μ] s := by rwa [Equiv.image_eq_preimage] at hs
state_before: case intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ ↑k).symm ⁻¹' s =ᵐ[μ] s
state_after: case intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℕ
hs : ⇑e⁻¹ ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ ↑k).symm ⁻¹' s =ᵐ[μ] s

tactic: replace he' : (⇑e⁻¹)^[k] ⁻¹' s =ᵐ[μ] s := he'.preimage_iterate_ae_eq k hs
state_before: case intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℕ
hs : ⇑e⁻¹ ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ ↑k).symm ⁻¹' s =ᵐ[μ] s
state_after: case intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
k : ℕ
hs : ⇑e⁻¹ ⁻¹' s =ᵐ[μ] s
he' : (⇑e⁻¹)^[k] ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ ↑k).symm ⁻¹' s =ᵐ[μ] s

tactic: rwa [Equiv.Perm.iterate_eq_pow e⁻¹ k, inv_pow e k] at he'
state_before: case intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
k : ℕ
hs : ⇑e⁻¹ ⁻¹' s =ᵐ[μ] s
he' : (⇑e⁻¹)^[k] ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ ↑k).symm ⁻¹' s =ᵐ[μ] s
state_after: no goals

tactic: rwa [Equiv.image_eq_preimage] at hs
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑e⁻¹ ⁻¹' s =ᵐ[μ] s
state_after: no goals

tactic: rw [zpow_neg, zpow_natCast]
state_before: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ (-↑k)).symm ⁻¹' s =ᵐ[μ] s
state_after: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s

tactic: replace hs : e ⁻¹' s =ᵐ[μ] s := by
  convert he.preimage_ae_eq hs.symm
  rw [Equiv.preimage_image]
state_before: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s
state_after: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℕ
hs : ⇑e ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s

tactic: replace he : (⇑e)^[k] ⁻¹' s =ᵐ[μ] s := he.preimage_iterate_ae_eq k hs
state_before: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℕ
hs : ⇑e ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s
state_after: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℕ
hs : ⇑e ⁻¹' s =ᵐ[μ] s
he : (⇑e)^[k] ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s

tactic: rwa [Equiv.Perm.iterate_eq_pow e k] at he
state_before: case intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
k : ℕ
hs : ⇑e ⁻¹' s =ᵐ[μ] s
he : (⇑e)^[k] ⁻¹' s =ᵐ[μ] s
⊢ ⇑(e ^ k)⁻¹.symm ⁻¹' s =ᵐ[μ] s
state_after: no goals

tactic: convert he.preimage_ae_eq hs.symm
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑e ⁻¹' s =ᵐ[μ] s
state_after: case h.e'_5
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ s = ⇑e ⁻¹' (⇑e '' s)

tactic: rw [Equiv.preimage_image]
state_before: case h.e'_5
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f : α → β
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ s = ⇑e ⁻¹' (⇑e '' s)
state_after: no goals