Theorem: measurableSet_eq_fun
File: Mathlib/MeasureTheory/Group/Arithmetic.lean

tactic: suffices h_set_eq : { x : α | f x = g x } = { x | (f - g) x = (0 : E) } by
  rw [h_set_eq]
  exact (hf.sub hg) measurableSet_eq
state_before: α✝ : Type u_1
G : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m✝ : MeasurableSpace α
f✝ g✝ : α → G
μ : Measure α
m : MeasurableSpace α
E : Type u_4
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
⊢ MeasurableSet {x | f x = g x}
state_after: α✝ : Type u_1
G : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m✝ : MeasurableSpace α
f✝ g✝ : α → G
μ : Measure α
m : MeasurableSpace α
E : Type u_4
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
⊢ {x | f x = g x} = {x | (f - g) x = 0}

tactic: ext
state_before: α✝ : Type u_1
G : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m✝ : MeasurableSpace α
f✝ g✝ : α → G
μ : Measure α
m : MeasurableSpace α
E : Type u_4
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
⊢ {x | f x = g x} = {x | (f - g) x = 0}
state_after: case h
α✝ : Type u_1
G : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m✝ : MeasurableSpace α
f✝ g✝ : α → G
μ : Measure α
m : MeasurableSpace α
E : Type u_4
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
x✝ : α
⊢ x✝ ∈ {x | f x = g x} ↔ x✝ ∈ {x | (f - g) x = 0}

tactic: simp_rw [Set.mem_setOf_eq, Pi.sub_apply, sub_eq_zero]
state_before: case h
α✝ : Type u_1
G : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m✝ : MeasurableSpace α
f✝ g✝ : α → G
μ : Measure α
m : MeasurableSpace α
E : Type u_4
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
x✝ : α
⊢ x✝ ∈ {x | f x = g x} ↔ x✝ ∈ {x | (f - g) x = 0}
state_after: no goals

tactic: rw [h_set_eq]
state_before: α✝ : Type u_1
G : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m✝ : MeasurableSpace α
f✝ g✝ : α → G
μ : Measure α
m : MeasurableSpace α
E : Type u_4
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
h_set_eq : {x | f x = g x} = {x | (f - g) x = 0}
⊢ MeasurableSet {x | f x = g x}
state_after: α✝ : Type u_1
G : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m✝ : MeasurableSpace α
f✝ g✝ : α → G
μ : Measure α
m : MeasurableSpace α
E : Type u_4
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
h_set_eq : {x | f x = g x} = {x | (f - g) x = 0}
⊢ MeasurableSet {x | (f - g) x = 0}

tactic: exact (hf.sub hg) measurableSet_eq
state_before: α✝ : Type u_1
G : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m✝ : MeasurableSpace α
f✝ g✝ : α → G
μ : Measure α
m : MeasurableSpace α
E : Type u_4
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
f g : α → E
hf : Measurable f
hg : Measurable g
h_set_eq : {x | f x = g x} = {x | (f - g) x = 0}
⊢ MeasurableSet {x | (f - g) x = 0}
state_after: no goals