Metamath Proof Explorer


Theorem xblcntr

Description: A ball contains its center. (Contributed by NM, 2-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Assertion xblcntr D ∞Met X P X R * 0 < R P P ball D R

Proof

Step Hyp Ref Expression
1 simp2 D ∞Met X P X R * 0 < R P X
2 xmet0 D ∞Met X P X P D P = 0
3 2 3adant3 D ∞Met X P X R * 0 < R P D P = 0
4 simp3r D ∞Met X P X R * 0 < R 0 < R
5 3 4 eqbrtrd D ∞Met X P X R * 0 < R P D P < R
6 elbl D ∞Met X P X R * P P ball D R P X P D P < R
7 6 3adant3r D ∞Met X P X R * 0 < R P P ball D R P X P D P < R
8 1 5 7 mpbir2and D ∞Met X P X R * 0 < R P P ball D R