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Theorem blcntrps 21364
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 11298 . . 3  |-  ( R  e.  RR+  ->  R  e. 
RR* )
2 rpgt0 11302 . . 3  |-  ( R  e.  RR+  ->  0  < 
R )
31, 2jca 534 . 2  |-  ( R  e.  RR+  ->  ( R  e.  RR*  /\  0  <  R ) )
4 xblcntrps 21362 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P
( ball `  D ) R ) )
53, 4syl3an3 1299 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 370    /\ w3a 982    e. wcel 1867   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   0cc0 9528   RR*cxr 9663    < clt 9664   RR+crp 11291  PsMetcpsmet 18895   ballcbl 18898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-map 7473  df-xr 9668  df-rp 11292  df-psmet 18903  df-bl 18906
This theorem is referenced by:  unirnblps  21371  blssexps  21378
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