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Theorem blval 18323
Description: The ball around a point  P is the set of all points whose distance from  P is less than the ball's radius  R. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Assertion
Ref Expression
blval  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  =  { x  e.  X  |  ( P D x )  < 
R } )
Distinct variable groups:    x, P    x, D    x, R    x, X

Proof of Theorem blval
Dummy variables  r 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blfval 18322 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D )  =  ( y  e.  X ,  r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
213ad2ant1 978 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ball `  D
)  =  ( y  e.  X ,  r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
3 simprl 733 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
y  =  P )
43oveq1d 6036 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( y D x )  =  ( P D x ) )
5 simprr 734 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
r  =  R )
64, 5breq12d 4167 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( ( y D x )  <  r  <->  ( P D x )  <  R ) )
76rabbidv 2892 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  ->  { x  e.  X  |  ( y D x )  <  r }  =  { x  e.  X  |  ( P D x )  < 
R } )
8 simp2 958 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  P  e.  X )
9 simp3 959 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  R  e.  RR* )
10 elfvdm 5698 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  X  e.  dom  * Met )
11103ad2ant1 978 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  X  e.  dom  * Met )
12 rabexg 4295 . . 3  |-  ( X  e.  dom  * Met  ->  { x  e.  X  |  ( P D x )  <  R }  e.  _V )
1311, 12syl 16 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  { x  e.  X  |  ( P D x )  <  R }  e.  _V )
142, 7, 8, 9, 13ovmpt2d 6141 1  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  =  { x  e.  X  |  ( P D x )  < 
R } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2654   _Vcvv 2900   class class class wbr 4154   dom cdm 4819   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   RR*cxr 9053    < clt 9054   * Metcxmt 16613   ballcbl 16615
This theorem is referenced by:  elbl  18324  metss2lem  18432  stdbdbl  18438  blval2  18483  nmhmcn  19000  lgamucov  24602  isbnd3  26185
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-map 6957  df-xr 9058  df-xmet 16620  df-bl 16622
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