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Metamath Proof Explorer

Theorem elbl2ps

Description: Membership in a ball. (Contributed by NM, 9-Mar-2007) (Revised by Thierry Arnoux, 11-Mar-2018)

Ref Expression
Assertion elbl2ps ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) )

Proof

Step Hyp Ref Expression
1 simprr ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → 𝐴 ∈ 𝑋 )
2 elblps ⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) ) )
3 2 3expa ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) ) )
4 3 an32s ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ 𝑃 ∈ 𝑋 ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) ) )
5 4 adantrr ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) ) )
6 1 5 mpbirand ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) )