Metamath Proof Explorer


Theorem xblcntrps

Description: A ball contains its center. (Contributed by NM, 2-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013) (Revised by Thierry Arnoux, 11-Mar-2018)

Ref Expression
Assertion xblcntrps DPsMetXPXR*0<RPPballDR

Proof

Step Hyp Ref Expression
1 simp2 DPsMetXPXR*0<RPX
2 psmet0 DPsMetXPXPDP=0
3 2 3adant3 DPsMetXPXR*0<RPDP=0
4 simp3r DPsMetXPXR*0<R0<R
5 3 4 eqbrtrd DPsMetXPXR*0<RPDP<R
6 elblps DPsMetXPXR*PPballDRPXPDP<R
7 6 3adant3r DPsMetXPXR*0<RPPballDRPXPDP<R
8 1 5 7 mpbir2and DPsMetXPXR*0<RPPballDR