Theorem xblcntrps 13053

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 988	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. X )
2		psmet0 12967	. . . 4 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( P D P ) = 0 )
3	2	3adant3 1007	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> ( P D P ) = 0 )
4		simp3r 1016	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> 0 < R )
5	3, 4	eqbrtrd 4004	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> ( P D P ) < R )
6		elblps 13030	. . 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 1225	. 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 934	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 968   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   0cc0 7753   RR*cxr 7932   < clt 7933  PsMetcpsmet 12619   ballcbl 12622

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-pnf 7935  df-mnf 7936  df-xr 7937  df-psmet 12627  df-bl 12630

This theorem is referenced by:  blcntrps  13055