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Theorem xblcntr 13547
Description: A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
xblcntr  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P ( ball `  D
) R ) )

Proof of Theorem xblcntr
StepHypRef Expression
1 simp2 998 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  X )
2 xmet0 13496 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  ( P D P )  =  0 )
323adant3 1017 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  ( P D P )  =  0 )
4 simp3r 1026 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  0  <  R )
53, 4eqbrtrd 4022 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  ( P D P )  <  R
)
6 elbl 13524 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P  e.  ( P ( ball `  D
) R )  <->  ( P  e.  X  /\  ( P D P )  < 
R ) ) )
763adant3r 1235 . 2  |-  ( ( D  e.  ( *Met `  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 944 1  |-  ( ( D  e.  ( *Met `  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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4000   ` cfv 5211  (class class class)co 5868   0cc0 7789   RR*cxr 7968    < clt 7969   *Metcxmet 13113   ballcbl 13115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-map 6643  df-pnf 7971  df-mnf 7972  df-xr 7973  df-psmet 13120  df-xmet 13121  df-bl 13123
This theorem is referenced by:  blcntr  13549  xblm  13550
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