Metamath Proof Explorer

Theorem elbl

Description: Membership in a ball. (Contributed by NM, 2-Sep-2006) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Assertion elbl ⊢ D ∈ ∞Met ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → A ∈ P ball ⁡ D R ↔ A ∈ X ∧ P D A < R

Proof

Step Hyp Ref Expression
1 blval ⊢ D ∈ ∞Met ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → P ball ⁡ D R = x ∈ X | P D x < R
2 1 eleq2d ⊢ D ∈ ∞Met ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → A ∈ P ball ⁡ D R ↔ A ∈ x ∈ X | P D x < R
3 oveq2 ⊢ x = A → P D x = P D A
4 3 breq1d ⊢ x = A → P D x < R ↔ P D A < R
5 4 elrab ⊢ A ∈ x ∈ X | P D x < R ↔ A ∈ X ∧ P D A < R
6 2 5 bitrdi ⊢ D ∈ ∞Met ⁡ X ∧ P ∈ X ∧ R ∈ ℝ * → A ∈ P ball ⁡ D R ↔ A ∈ X ∧ P D A < R