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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 ) ) )