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

Theorem blssec 18348
Description: A ball centered at  P is contained in the set of points finitely separated from  P. This is just an application of ssbl 18338 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
Hypothesis
Ref Expression
xmeter.1  |-  .~  =  ( `' D " RR )
Assertion
Ref Expression
blssec  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( P ( ball `  D ) S ) 
C_  [ P ]  .~  )

Proof of Theorem blssec
StepHypRef Expression
1 pnfge 10652 . . . . 5  |-  ( S  e.  RR*  ->  S  <_  +oo )
21adantl 453 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  S  e.  RR* )  ->  S  <_  +oo )
3 pnfxr 10638 . . . . 5  |-  +oo  e.  RR*
4 ssbl 18338 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( S  e.  RR*  /\  +oo  e.  RR* )  /\  S  <_  +oo )  ->  ( P ( ball `  D
) S )  C_  ( P ( ball `  D
)  +oo ) )
543expia 1155 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( S  e.  RR*  /\  +oo  e.  RR* ) )  -> 
( S  <_  +oo  ->  ( P ( ball `  D
) S )  C_  ( P ( ball `  D
)  +oo ) ) )
63, 5mpanr2 666 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  S  e.  RR* )  ->  ( S  <_  +oo  ->  ( P ( ball `  D
) S )  C_  ( P ( ball `  D
)  +oo ) ) )
72, 6mpd 15 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  S  e.  RR* )  ->  ( P ( ball `  D
) S )  C_  ( P ( ball `  D
)  +oo ) )
873impa 1148 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( P ( ball `  D ) S ) 
C_  ( P (
ball `  D )  +oo ) )
9 xmeter.1 . . . 4  |-  .~  =  ( `' D " RR )
109xmetec 18347 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  [ P ]  .~  =  ( P ( ball `  D
)  +oo ) )
11103adant3 977 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  [ P ]  .~  =  ( P (
ball `  D )  +oo ) )
128, 11sseqtr4d 3321 1  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( P ( ball `  D ) S ) 
C_  [ P ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3256   class class class wbr 4146   `'ccnv 4810   "cima 4814   ` cfv 5387  (class class class)co 6013   [cec 6832   RRcr 8915    +oocpnf 9043   RR*cxr 9045    <_ cle 9047   * Metcxmt 16605   ballcbl 16607
This theorem is referenced by:  xmetresbl  18350  xrsblre  18706  isbndx  26175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-er 6834  df-ec 6836  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-2 9983  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-xmet 16612  df-bl 16614
  Copyright terms: Public domain W3C validator