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Theorem blvalps 21331
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 21329 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( y  e.  X , 
r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
213ad2ant1 1026 . 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 762 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  y  =  P )
43oveq1d 6320 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  ( y D x )  =  ( P D x ) )
5 simprr 764 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  r  =  R )
64, 5breq12d 4439 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  ( (
y D x )  <  r  <->  ( P D x )  < 
R ) )
76rabbidv 3079 . 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 1006 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  P  e.  X )
9 simp3 1007 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  R  e.  RR* )
10 elfvdm 5907 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
11103ad2ant1 1026 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  X  e.  dom PsMet )
12 rabexg 4575 . . 3  |-  ( X  e.  dom PsMet  ->  { x  e.  X  |  ( P D x )  < 
R }  e.  _V )
1311, 12syl 17 . 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 6438 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087   class class class wbr 4426   dom cdm 4854   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   RR*cxr 9673    < clt 9674  PsMetcpsmet 18889   ballcbl 18892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-xr 9678  df-psmet 18897  df-bl 18900
This theorem is referenced by:  elblps  21333  blval2  21508
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