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Theorem elbl2ps 20977
Description: Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
elbl2ps  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X )
)  ->  ( A  e.  ( P ( ball `  D ) R )  <-> 
( P D A )  <  R ) )

Proof of Theorem elbl2ps
StepHypRef Expression
1 elblps 20975 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( A  e.  ( P
( ball `  D ) R )  <->  ( A  e.  X  /\  ( P D A )  < 
R ) ) )
213expa 1194 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  R  e.  RR* )  ->  ( A  e.  ( P
( ball `  D ) R )  <->  ( A  e.  X  /\  ( P D A )  < 
R ) ) )
32an32s 802 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR* )  /\  P  e.  X )  ->  ( A  e.  ( P
( ball `  D ) R )  <->  ( A  e.  X  /\  ( P D A )  < 
R ) ) )
43adantrr 714 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X )
)  ->  ( A  e.  ( P ( ball `  D ) R )  <-> 
( A  e.  X  /\  ( P D A )  <  R ) ) )
5 simprr 755 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X )
)  ->  A  e.  X )
65biantrurd 506 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X )
)  ->  ( ( P D A )  < 
R  <->  ( A  e.  X  /\  ( P D A )  < 
R ) ) )
74, 6bitr4d 256 1  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR* )  /\  ( P  e.  X  /\  A  e.  X )
)  ->  ( A  e.  ( P ( ball `  D ) R )  <-> 
( P D A )  <  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1826   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   RR*cxr 9538    < clt 9539  PsMetcpsmet 18515   ballcbl 18518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-map 7340  df-xr 9543  df-psmet 18524  df-bl 18527
This theorem is referenced by:  elbl3ps  20979  blcomps  20981
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