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Theorem blcntrps 20742
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
blcntrps  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  P  e.  ( P ( ball `  D ) R ) )

Proof of Theorem blcntrps
StepHypRef Expression
1 rpxr 11228 . . 3  |-  ( R  e.  RR+  ->  R  e. 
RR* )
2 rpgt0 11232 . . 3  |-  ( R  e.  RR+  ->  0  < 
R )
31, 2jca 532 . 2  |-  ( R  e.  RR+  ->  ( R  e.  RR*  /\  0  <  R ) )
4 xblcntrps 20740 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P
( ball `  D ) R ) )
53, 4syl3an3 1263 1  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  P  e.  ( P ( ball `  D ) R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   0cc0 9493   RR*cxr 9628    < clt 9629   RR+crp 11221  PsMetcpsmet 18213   ballcbl 18216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-map 7423  df-xr 9633  df-rp 11222  df-psmet 18222  df-bl 18225
This theorem is referenced by:  unirnblps  20749  blssexps  20756
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