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Theorem blvalps 20756
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.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blvalps  |-  ( ( D  e.  (PsMet `  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 blvalps
Dummy variables  r 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blfvalps 20754 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( y  e.  X , 
r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
213ad2ant1 1017 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( ball `  D )  =  ( y  e.  X ,  r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
3 simprl 755 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  y  =  P )
43oveq1d 6310 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  ( y D x )  =  ( P D x ) )
5 simprr 756 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  r  =  R )
64, 5breq12d 4466 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  ( (
y D x )  <  r  <->  ( P D x )  < 
R ) )
76rabbidv 3110 . 2  |-  ( ( ( D  e.  (PsMet `  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 997 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  P  e.  X )
9 simp3 998 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  R  e.  RR* )
10 elfvdm 5898 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
11103ad2ant1 1017 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  X  e.  dom PsMet )
12 rabexg 4603 . . 3  |-  ( X  e.  dom PsMet  ->  { x  e.  X  |  ( P D x )  < 
R }  e.  _V )
1311, 12syl 16 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  { x  e.  X  |  ( P D x )  < 
R }  e.  _V )
142, 7, 8, 9, 13ovmpt2d 6425 1  |-  ( ( D  e.  (PsMet `  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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118   class class class wbr 4453   dom cdm 5005   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   RR*cxr 9639    < clt 9640  PsMetcpsmet 18272   ballcbl 18275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-xr 9644  df-psmet 18281  df-bl 18284
This theorem is referenced by:  elblps  20758  blval2  20946
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