Metamath Proof Explorer


Theorem blnei

Description: A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007) (Revised by Mario Carneiro, 24-Aug-2015)

Ref Expression
Hypothesis mopni.1 J = MetOpen D
Assertion blnei D ∞Met X P X R + P ball D R nei J P

Proof

Step Hyp Ref Expression
1 mopni.1 J = MetOpen D
2 1 mopntop D ∞Met X J Top
3 2 3ad2ant1 D ∞Met X P X R + J Top
4 rpxr R + R *
5 1 blopn D ∞Met X P X R * P ball D R J
6 4 5 syl3an3 D ∞Met X P X R + P ball D R J
7 blcntr D ∞Met X P X R + P P ball D R
8 opnneip J Top P ball D R J P P ball D R P ball D R nei J P
9 3 6 7 8 syl3anc D ∞Met X P X R + P ball D R nei J P