Theorem xblcntrps 12800

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.)

Assertion

Ref	Expression
xblcntrps	|- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. ( P ( ball ` D ) R ) )

Proof of Theorem xblcntrps

Step	Hyp	Ref	Expression
1		simp2 983	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. X )
2		psmet0 12714	. . . 4 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( P D P ) = 0 )
3	2	3adant3 1002	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> ( P D P ) = 0 )
4		simp3r 1011	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> 0 < R )
5	3, 4	eqbrtrd 3986	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> ( P D P ) < R )
6		elblps 12777	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( P e. ( P ( ball ` D ) R ) <-> ( P e. X /\ ( P D P ) < R ) ) )
7	6	3adant3r 1217	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> ( P e. ( P ( ball ` D ) R ) <-> ( P e. X /\ ( P D P ) < R ) ) )
8	1, 5, 7	mpbir2and 929	1 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. ( P ( ball ` D ) R ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1335   e. wcel 2128   class class class wbr 3965   ` cfv 5169  (class class class)co 5821   0cc0 7727   RR*cxr 7906   < clt 7907  PsMetcpsmet 12366   ballcbl 12369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7818  ax-resscn 7819

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-map 6592  df-pnf 7909  df-mnf 7910  df-xr 7911  df-psmet 12374  df-bl 12377

This theorem is referenced by:  blcntrps  12802