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Theorem blvalps 20093
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 20091 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( y  e.  X , 
r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
213ad2ant1 1009 . 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 6216 . . . 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 4414 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  /\  (
y  =  P  /\  r  =  R )
)  ->  ( (
y D x )  <  r  <->  ( P D x )  < 
R ) )
76rabbidv 3070 . 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 989 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  P  e.  X )
9 simp3 990 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  R  e.  RR* )
10 elfvdm 5826 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
11103ad2ant1 1009 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  X  e.  dom PsMet )
12 rabexg 4551 . . 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 6329 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 965    = wceq 1370    e. wcel 1758   {crab 2803   _Vcvv 3078   class class class wbr 4401   dom cdm 4949   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   RR*cxr 9529    < clt 9530  PsMetcpsmet 17926   ballcbl 17929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327  df-xr 9534  df-psmet 17935  df-bl 17938
This theorem is referenced by:  elblps  20095  blval2  20283
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