MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  blval Unicode version

Theorem blval 17964
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 17963 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D )  =  ( y  e.  X ,  r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
213ad2ant1 976 . 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 732 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
y  =  P )
43oveq1d 5889 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( y D x )  =  ( P D x ) )
5 simprr 733 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
r  =  R )
64, 5breq12d 4052 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( ( y D x )  <  r  <->  ( P D x )  <  R ) )
76rabbidv 2793 . 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 956 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  P  e.  X )
9 simp3 957 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  R  e.  RR* )
10 elfvdm 5570 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  X  e.  dom  * Met )
11103ad2ant1 976 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  X  e.  dom  * Met )
12 rabexg 4180 . . 3  |-  ( X  e.  dom  * Met  ->  { x  e.  X  |  ( P D x )  <  R }  e.  _V )
1311, 12syl 15 . 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 5991 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   class class class wbr 4039   dom cdm 4705   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   RR*cxr 8882    < clt 8883   * Metcxmt 16385   ballcbl 16387
This theorem is referenced by:  elbl  17965  metss2lem  18073  stdbdbl  18079  nmhmcn  18617  isbnd3  26611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-xr 8887  df-xmet 16389  df-bl 16391
  Copyright terms: Public domain W3C validator