Theorem: MeasureTheory.IsFundamentalDomain.quotientMeasure_eq
File: Mathlib/MeasureTheory/Group/FundamentalDomain.lean

Proof steps:
state_before: G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
s t : Set α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : MeasurableSMul G α
fund_dom_s : IsFundamentalDomain G s μ
fund_dom_t : IsFundamentalDomain G t μ
⊢ Measure.map (Quotient.mk α_mod_G) (μ.restrict s) = Measure.map (Quotient.mk α_mod_G) (μ.restrict t)
state_after: case h
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
s t : Set α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : MeasurableSMul G α
fund_dom_s : IsFundamentalDomain G s μ
fund_dom_t : IsFundamentalDomain G t μ
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U
⊢ (Measure.map (Quotient.mk α_mod_G) (μ.restrict s)) U = (Measure.map (Quotient.mk α_mod_G) (μ.restrict t)) U
tactic: ext U meas_U
state_before: case h
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
s t : Set α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : MeasurableSMul G α
fund_dom_s : IsFundamentalDomain G s μ
fund_dom_t : IsFundamentalDomain G t μ
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U
⊢ (Measure.map (Quotient.mk α_mod_G) (μ.restrict s)) U = (Measure.map (Quotient.mk α_mod_G) (μ.restrict t)) U
state_after: case h
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
s t : Set α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : MeasurableSMul G α
fund_dom_s : IsFundamentalDomain G s μ
fund_dom_t : IsFundamentalDomain G t μ
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U
⊢ μ (Quotient.mk α_mod_G ⁻¹' U ∩ s) = μ (Quotient.mk α_mod_G ⁻¹' U ∩ t)
tactic: rw [measure_map_restrict_apply (meas_U := meas_U), measure_map_restrict_apply (meas_U := meas_U)]
state_before: case h
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
s t : Set α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : MeasurableSMul G α
fund_dom_s : IsFundamentalDomain G s μ
fund_dom_t : IsFundamentalDomain G t μ
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U
⊢ μ (Quotient.mk α_mod_G ⁻¹' U ∩ s) = μ (Quotient.mk α_mod_G ⁻¹' U ∩ t)
state_after: case h.hA₀
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
s t : Set α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : MeasurableSMul G α
fund_dom_s : IsFundamentalDomain G s μ
fund_dom_t : IsFundamentalDomain G t μ
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U
⊢ MeasurableSet (Quotient.mk α_mod_G ⁻¹' U)

case h.hA
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : Countable G
inst✝² : MeasurableSpace G
s t : Set α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : MeasurableSMul G α
fund_dom_s : IsFundamentalDomain G s μ
fund_dom_t : IsFundamentalDomain G t μ
U : Set (Quotient α_mod_G)
meas_U : MeasurableSet U
⊢ ∀ (g : G), (fun x => g • x) ⁻¹' (Quotient.mk α_mod_G ⁻¹' U) = Quotient.mk α_mod_G ⁻¹' U
tactic: apply MeasureTheory.IsFundamentalDomain.measure_set_eq fund_dom_s fund_dom_t