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Theorem xblcntrps 12571
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
StepHypRef Expression
1 simp2 982 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  X )
2 psmet0 12485 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( P D P )  =  0 )
323adant3 1001 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  -> 
( P D P )  =  0 )
4 simp3r 1010 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  -> 
0  <  R )
53, 4eqbrtrd 3945 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  -> 
( P D P )  <  R )
6 elblps 12548 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P  e.  ( P
( ball `  D ) R )  <->  ( P  e.  X  /\  ( P D P )  < 
R ) ) )
763adant3r 1213 . 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 ) ) )
81, 5, 7mpbir2and 928 1  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P
( ball `  D ) R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   0cc0 7613   RR*cxr 7792    < clt 7793  PsMetcpsmet 12137   ballcbl 12140
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-map 6537  df-pnf 7795  df-mnf 7796  df-xr 7797  df-psmet 12145  df-bl 12148
This theorem is referenced by:  blcntrps  12573
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