Theorem: sphere_pi
File: Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean

tactic: obtain hr | rfl | hr := lt_trichotomy r 0
state_before: α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
r : ℝ
h : 0 < r ∨ Nonempty β
⊢ sphere x r = (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r
state_after: case inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
r : ℝ
h : 0 < r ∨ Nonempty β
hr : r < 0
⊢ sphere x r = (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r

case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
⊢ sphere x 0 = (⋃ i, Function.eval i ⁻¹' sphere (x i) 0) ∩ closedBall x 0

case inr.inr
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
r : ℝ
h : 0 < r ∨ Nonempty β
hr : 0 < r
⊢ sphere x r = (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r

tactic: simp [hr]
state_before: case inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
r : ℝ
h : 0 < r ∨ Nonempty β
hr : r < 0
⊢ sphere x r = (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r
state_after: no goals

tactic: rw [closedBall_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right]
state_before: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
⊢ sphere x 0 = (⋃ i, Function.eval i ⁻¹' sphere (x i) 0) ∩ closedBall x 0
state_after: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
⊢ sphere x 0 ⊆ ⋃ i, Function.eval i ⁻¹' sphere (x i) 0

tactic: letI := h.resolve_left (lt_irrefl _)
state_before: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
⊢ sphere x 0 ⊆ ⋃ i, Function.eval i ⁻¹' sphere (x i) 0
state_after: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
⊢ sphere x 0 ⊆ ⋃ i, Function.eval i ⁻¹' sphere (x i) 0

tactic: inhabit β
state_before: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
⊢ sphere x 0 ⊆ ⋃ i, Function.eval i ⁻¹' sphere (x i) 0
state_after: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
⊢ sphere x 0 ⊆ ⋃ i, Function.eval i ⁻¹' sphere (x i) 0

tactic: refine subset_iUnion_of_subset default ?_
state_before: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
⊢ sphere x 0 ⊆ ⋃ i, Function.eval i ⁻¹' sphere (x i) 0
state_after: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
⊢ sphere x 0 ⊆ Function.eval default ⁻¹' sphere (x default) 0

tactic: intro x hx
state_before: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
⊢ sphere x 0 ⊆ Function.eval default ⁻¹' sphere (x default) 0
state_after: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x✝ : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
x : (x : β) → π x
hx : x ∈ sphere x✝ 0
⊢ x ∈ Function.eval default ⁻¹' sphere (x✝ default) 0

tactic: replace hx := hx.le
state_before: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x✝ : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
x : (x : β) → π x
hx : x ∈ sphere x✝ 0
⊢ x ∈ Function.eval default ⁻¹' sphere (x✝ default) 0
state_after: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x✝ : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
x : (x : β) → π x
hx : dist x x✝ ≤ 0
⊢ x ∈ Function.eval default ⁻¹' sphere (x✝ default) 0

tactic: rw [dist_pi_le_iff le_rfl] at hx
state_before: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x✝ : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
x : (x : β) → π x
hx : dist x x✝ ≤ 0
⊢ x ∈ Function.eval default ⁻¹' sphere (x✝ default) 0
state_after: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x✝ : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
x : (x : β) → π x
hx : ∀ (b : β), dist (x b) (x✝ b) ≤ 0
⊢ x ∈ Function.eval default ⁻¹' sphere (x✝ default) 0

tactic: exact le_antisymm (hx default) dist_nonneg
state_before: case inr.inl
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x✝ : (b : β) → π b
h : 0 < 0 ∨ Nonempty β
this : Nonempty β := Or.resolve_left h (lt_irrefl 0)
inhabited_h : Inhabited β
x : (x : β) → π x
hx : ∀ (b : β), dist (x b) (x✝ b) ≤ 0
⊢ x ∈ Function.eval default ⁻¹' sphere (x✝ default) 0
state_after: no goals

tactic: ext
state_before: case inr.inr
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
r : ℝ
h : 0 < r ∨ Nonempty β
hr : 0 < r
⊢ sphere x r = (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r
state_after: case inr.inr.h
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
r : ℝ
h : 0 < r ∨ Nonempty β
hr : 0 < r
x✝ : (b : β) → π b
⊢ x✝ ∈ sphere x r ↔ x✝ ∈ (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r

tactic: simp [dist_pi_eq_iff hr, dist_pi_le_iff hr.le]
state_before: case inr.inr.h
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
x : (b : β) → π b
r : ℝ
h : 0 < r ∨ Nonempty β
hr : 0 < r
x✝ : (b : β) → π b
⊢ x✝ ∈ sphere x r ↔ x✝ ∈ (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r
state_after: no goals