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Theorem blvalps 12546
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 12543 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( y  e.  X , 
r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
213ad2ant1 1002 . 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 520 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  y  =  P )
43oveq1d 5782 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  ( y D x )  =  ( P D x ) )
5 simprr 521 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  r  =  R )
64, 5breq12d 3937 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  ( (
y D x )  <  r  <->  ( P D x )  < 
R ) )
76rabbidv 2670 . 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 982 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  P  e.  X )
9 simp3 983 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  R  e.  RR* )
10 psmetrel 12480 . . . . 5  |-  Rel PsMet
11 relelfvdm 5446 . . . . 5  |-  ( ( Rel PsMet  /\  D  e.  (PsMet `  X ) )  ->  X  e.  dom PsMet )
1210, 11mpan 420 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
13123ad2ant1 1002 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  X  e.  dom PsMet )
14 rabexg 4066 . . 3  |-  ( X  e.  dom PsMet  ->  { x  e.  X  |  ( P D x )  < 
R }  e.  _V )
1513, 14syl 14 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  { x  e.  X  |  ( P D x )  < 
R }  e.  _V )
162, 7, 8, 9, 15ovmpod 5891 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 set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   {crab 2418   _Vcvv 2681   class class class wbr 3924   dom cdm 4534   Rel wrel 4539   ` cfv 5118  (class class class)co 5767    e. cmpo 5769   RR*cxr 7792    < clt 7793  PsMetcpsmet 12137   ballcbl 12140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-map 6537  df-pnf 7795  df-mnf 7796  df-xr 7797  df-psmet 12145  df-bl 12148
This theorem is referenced by:  elblps  12548
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