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Metamath Proof Explorer

Theorem blcomps

Description: Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014) (Revised by Thierry Arnoux, 11-Mar-2018)

Ref Expression
Assertion blcomps ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ 𝑃 ∈ ( 𝐴 ( ball ‘ 𝐷 ) 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 elbl3ps ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋 ) ) → ( 𝑃 ∈ ( 𝐴 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) )
3 2 ancom2s ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝑃 ∈ ( 𝐴 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) )
4 1 3 bitr4d ⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ) ∧ ( 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ 𝑃 ∈ ( 𝐴 ( ball ‘ 𝐷 ) 𝑅 ) ) )