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Theorem blssec 17997
Description: A ball centered at  P is contained in the set of points finitely separated from  P. This is just an application of ssbl 17987 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 10485 . . . . 5  |-  ( S  e.  RR*  ->  S  <_  +oo )
21adantl 452 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  S  e.  RR* )  ->  S  <_  +oo )
3 pnfxr 10471 . . . . 5  |-  +oo  e.  RR*
4 ssbl 17987 . . . . . 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 1153 . . . . 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 665 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  S  e.  RR* )  ->  ( S  <_  +oo  ->  ( P ( ball `  D
) S )  C_  ( P ( ball `  D
)  +oo ) ) )
72, 6mpd 14 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  S  e.  RR* )  ->  ( P ( ball `  D
) S )  C_  ( P ( ball `  D
)  +oo ) )
873impa 1146 . 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 17996 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  [ P ]  .~  =  ( P ( ball `  D
)  +oo ) )
11103adant3 975 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  [ P ]  .~  =  ( P (
ball `  D )  +oo ) )
128, 11sseqtr4d 3228 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874   [cec 6674   RRcr 8752    +oocpnf 8880   RR*cxr 8882    <_ cle 8884   * Metcxmt 16385   ballcbl 16387
This theorem is referenced by:  xmetresbl  17999  xrsblre  18333  isbndx  26609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-ec 6678  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-xmet 16389  df-bl 16391
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