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Theorem xblcntrps 18393
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 958 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  X )
2 psmet0 18292 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( P D P )  =  0 )
323adant3 977 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  -> 
( P D P )  =  0 )
4 simp3r 986 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  -> 
0  <  R )
53, 4eqbrtrd 4192 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  -> 
( P D P )  <  R )
6 elblps 18370 . . 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 1181 . 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 889 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   0cc0 8946   RR*cxr 9075    < clt 9076  PsMetcpsmet 16640   ballcbl 16643
This theorem is referenced by:  blcntrps  18395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-map 6979  df-xr 9080  df-psmet 16649  df-bl 16652
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