MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elbl Structured version   Unicode version

Theorem elbl 21057
Description: Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Assertion
Ref Expression
elbl  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( A  e.  ( P ( ball `  D
) R )  <->  ( A  e.  X  /\  ( P D A )  < 
R ) ) )

Proof of Theorem elbl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 blval 21055 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  =  { x  e.  X  |  ( P D x )  < 
R } )
21eleq2d 2524 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( A  e.  ( P ( ball `  D
) R )  <->  A  e.  { x  e.  X  | 
( P D x )  <  R }
) )
3 oveq2 6278 . . . 4  |-  ( x  =  A  ->  ( P D x )  =  ( P D A ) )
43breq1d 4449 . . 3  |-  ( x  =  A  ->  (
( P D x )  <  R  <->  ( P D A )  <  R
) )
54elrab 3254 . 2  |-  ( A  e.  { x  e.  X  |  ( P D x )  < 
R }  <->  ( A  e.  X  /\  ( P D A )  < 
R ) )
62, 5syl6bb 261 1  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( A  e.  ( P ( ball `  D
) R )  <->  ( A  e.  X  /\  ( P D A )  < 
R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {crab 2808   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RR*cxr 9616    < clt 9617   *Metcxmt 18598   ballcbl 18600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-xr 9621  df-psmet 18606  df-xmet 18607  df-bl 18609
This theorem is referenced by:  elbl2  21059  xblpnf  21065  bldisj  21067  blgt0  21068  xblss2  21071  blhalf  21074  xblcntr  21080  xbln0  21083  blin  21090  blss  21094  blres  21100  imasf1obl  21157  prdsbl  21160  blcls  21175  metcnp  21210  dscopn  21260  cnbl0  21447  bl2ioo  21463  blcvx  21469  xrsmopn  21483  recld2  21485  cnheibor  21621  nmhmcn  21769  lmmbr2  21864  iscau2  21882  dvlip2  22562  psercn  22987  abelth  23002  logtayl  23209  logtayl2  23211  heicant  30289  iooabslt  31771  limcrecl  31874  islpcn  31884
  Copyright terms: Public domain W3C validator