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Theorem xblm 14270
Description: A ball is inhabited iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
xblm  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( E. x  x  e.  ( P (
ball `  D ) R )  <->  0  <  R ) )
Distinct variable groups:    x, D    x, R    x, P    x, X

Proof of Theorem xblm
StepHypRef Expression
1 elbl 14244 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
2 xmetge0 14218 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  0  <_  ( P D x ) )
323expa 1204 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  0  <_  ( P D x ) )
433adantl3 1156 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  0  <_  ( P D x ) )
5 0xr 8018 . . . . . . 7  |-  0  e.  RR*
6 xmetcl 14205 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
763expa 1204 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  ( P D x )  e. 
RR* )
873adantl3 1156 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
9 simpl3 1003 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  R  e.  RR* )
10 xrlelttr 9820 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  ( P D x )  e. 
RR*  /\  R  e.  RR* )  ->  ( (
0  <_  ( P D x )  /\  ( P D x )  <  R )  -> 
0  <  R )
)
115, 8, 9, 10mp3an2i 1352 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( (
0  <_  ( P D x )  /\  ( P D x )  <  R )  -> 
0  <  R )
)
124, 11mpand 429 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( ( P D x )  < 
R  ->  0  <  R ) )
1312expimpd 363 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( x  e.  X  /\  ( P D x )  < 
R )  ->  0  <  R ) )
141, 13sylbid 150 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) R )  -> 
0  <  R )
)
1514exlimdv 1829 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( E. x  x  e.  ( P (
ball `  D ) R )  ->  0  <  R ) )
16 simpl2 1002 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  0  <  R
)  ->  P  e.  X )
17 simpl1 1001 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  0  <  R
)  ->  D  e.  ( *Met `  X
) )
18 simpl3 1003 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  0  <  R
)  ->  R  e.  RR* )
19 simpr 110 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  0  <  R
)  ->  0  <  R )
20 xblcntr 14267 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P ( ball `  D
) R ) )
2117, 16, 18, 19, 20syl112anc 1252 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  0  <  R
)  ->  P  e.  ( P ( ball `  D
) R ) )
22 eleq1 2250 . . . . 5  |-  ( x  =  P  ->  (
x  e.  ( P ( ball `  D
) R )  <->  P  e.  ( P ( ball `  D
) R ) ) )
2322spcegv 2837 . . . 4  |-  ( P  e.  X  ->  ( P  e.  ( P
( ball `  D ) R )  ->  E. x  x  e.  ( P
( ball `  D ) R ) ) )
2416, 21, 23sylc 62 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  0  <  R
)  ->  E. x  x  e.  ( P
( ball `  D ) R ) )
2524ex 115 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( 0  <  R  ->  E. x  x  e.  ( P ( ball `  D ) R ) ) )
2615, 25impbid 129 1  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( E. x  x  e.  ( P (
ball `  D ) R )  <->  0  <  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 979   E.wex 1502    e. wcel 2158   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   0cc0 7825   RR*cxr 8005    < clt 8006    <_ cle 8007   *Metcxmet 13779   ballcbl 13781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-ltadd 7941  ax-pre-mulgt0 7942
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-map 6664  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-2 8992  df-xadd 9787  df-psmet 13786  df-xmet 13787  df-bl 13789
This theorem is referenced by:  blssioo  14398
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