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Metamath Proof Explorer

Theorem elbl3ps

Description: Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007)

Ref Expression
Assertion elbl3ps ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝐴 𝐷 𝑃 ) < 𝑅 ) )

Proof

Step Hyp Ref Expression
1 elbl2ps ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) )
2 psmetsym ⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝑃 𝐷 𝐴 ) = ( 𝐴 𝐷 𝑃 ) )
3 2 3expb ⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝑃 𝐷 𝐴 ) = ( 𝐴 𝐷 𝑃 ) )
4 3 adantlr ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝑃 𝐷 𝐴 ) = ( 𝐴 𝐷 𝑃 ) )
5 4 breq1d ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝑃 𝐷 𝐴 ) < 𝑅 ↔ ( 𝐴 𝐷 𝑃 ) < 𝑅 ) )
6 1 5 bitrd ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝐴 𝐷 𝑃 ) < 𝑅 ) )