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Metamath Proof Explorer

Theorem blgt0

Description: A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007) (Revised by Mario Carneiro, 23-Aug-2015)

		Ref	Expression
	Assertion	blgt0	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → 0 < 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		0xr	⊢ 0 ∈ ℝ^*
2	1	a1i	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → 0 ∈ ℝ^* )
3		simpl1	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
4		simpl2	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → 𝑃 ∈ 𝑋 )
5		elbl	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) → ( 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝐴 ) < 𝑅 ) ) )
6	5	simprbda	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → 𝐴 ∈ 𝑋 )
7		xmetcl	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝑃 𝐷 𝐴 ) ∈ ℝ^* )
8	3 4 6 7	syl3anc	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → ( 𝑃 𝐷 𝐴 ) ∈ ℝ^* )
9		simpl3	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → 𝑅 ∈ ℝ^* )
10		xmetge0	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → 0 ≤ ( 𝑃 𝐷 𝐴 ) )
11	3 4 6 10	syl3anc	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → 0 ≤ ( 𝑃 𝐷 𝐴 ) )
12	5	simplbda	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → ( 𝑃 𝐷 𝐴 ) < 𝑅 )
13	2 8 9 11 12	xrlelttrd	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^* ) ∧ 𝐴 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → 0 < 𝑅 )