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Theorem blval 21058
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 21056 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D )  =  ( y  e.  X ,  r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
213ad2ant1 1015 . 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 754 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
y  =  P )
43oveq1d 6285 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( y D x )  =  ( P D x ) )
5 simprr 755 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
r  =  R )
64, 5breq12d 4452 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( ( y D x )  <  r  <->  ( P D x )  <  R ) )
76rabbidv 3098 . 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 995 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  P  e.  X )
9 simp3 996 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  R  e.  RR* )
10 elfvdm 5874 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
11103ad2ant1 1015 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  X  e.  dom  *Met )
12 rabexg 4587 . . 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 6403 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106   class class class wbr 4439   dom cdm 4988   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   RR*cxr 9616    < clt 9617   *Metcxmt 18601   ballcbl 18603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-xr 9621  df-psmet 18609  df-xmet 18610  df-bl 18612
This theorem is referenced by:  elbl  21060  metss2lem  21183  stdbdbl  21189  nmhmcn  21772  lgamucov  28847  isbnd3  30523
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