ι : Type u_2
α : ι → Type u_1
inst✝¹ : (i : ι) → Preorder (α i)
x✝ y✝ : (i : ι) → α i
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
hm : x i₀ ≤ m
this : Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀)
⊢ (pi univ fun i => Ioc (update x i₀ m i) (y i)) = {z | m < z i₀} ∩ pi univ fun i => Ioc (x i) (y i)
ι : Type u_2
α : ι → Type u_1
inst✝¹ : (i : ι) → Preorder (α i)
x✝ y✝ : (i : ι) → α i
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
hm : x i₀ ≤ m
⊢ (pi univ fun i => Ioc (update x i₀ m i) (y i)) = {z | m < z i₀} ∩ pi univ fun i => Ioc (x i) (y i)
have : Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀) := by
  rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, ← inter_assoc,
    inter_eq_self_of_subset_left (Ioi_subset_Ioi hm)]
ι : Type u_2
α : ι → Type u_1
inst✝¹ : (i : ι) → Preorder (α i)
x✝ y✝ : (i : ι) → α i
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
hm : x i₀ ≤ m
this : Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀)
⊢ ({x_1 | x_1 i₀ ∈ Ioi m ∩ Ioc (x i₀) (y i₀)} ∩ pi ({i₀}ᶜ) fun j => Ioc (x j) (y j)) =
    {z | m < z i₀} ∩ {x_1 | x_1 i₀ ∈ Ioc (x i₀) (y i₀)} ∩ pi ({i₀}ᶜ) fun j => Ioc (x j) (y j)
ι : Type u_2
α : ι → Type u_1
inst✝¹ : (i : ι) → Preorder (α i)
x✝ y✝ : (i : ι) → α i
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
hm : x i₀ ≤ m
this : Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀)
⊢ (pi univ fun i => Ioc (update x i₀ m i) (y i)) = {z | m < z i₀} ∩ pi univ fun i => Ioc (x i) (y i)
simp_rw [univ_pi_update i₀ _ _ fun i z ↦ Ioc z (y i), ← pi_inter_compl ({i₀} : Set ι),
  singleton_pi', ← inter_assoc, this]
no goals
ι : Type u_2
α : ι → Type u_1
inst✝¹ : (i : ι) → Preorder (α i)
x✝ y✝ : (i : ι) → α i
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
hm : x i₀ ≤ m
this : Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀)
⊢ ({x_1 | x_1 i₀ ∈ Ioi m ∩ Ioc (x i₀) (y i₀)} ∩ pi ({i₀}ᶜ) fun j => Ioc (x j) (y j)) =
    {z | m < z i₀} ∩ {x_1 | x_1 i₀ ∈ Ioc (x i₀) (y i₀)} ∩ pi ({i₀}ᶜ) fun j => Ioc (x j) (y j)
rfl
no goals
ι : Type u_2
α : ι → Type u_1
inst✝¹ : (i : ι) → Preorder (α i)
x✝ y✝ : (i : ι) → α i
inst✝ : DecidableEq ι
x y : (i : ι) → α i
i₀ : ι
m : α i₀
hm : x i₀ ≤ m
⊢ Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀)
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, ← inter_assoc,
  inter_eq_self_of_subset_left (Ioi_subset_Ioi hm)]