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Theorem xbln0 21353
Description: A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
xbln0  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( P (
ball `  D ) R )  =/=  (/)  <->  0  <  R ) )

Proof of Theorem xbln0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 n0 3768 . . 3  |-  ( ( P ( ball `  D
) R )  =/=  (/) 
<->  E. x  x  e.  ( P ( ball `  D ) R ) )
2 elbl 21327 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
3 xmetge0 21283 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  0  <_  ( P D x ) )
433expa 1205 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  0  <_  ( P D x ) )
543adantl3 1163 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  0  <_  ( P D x ) )
6 0xr 9676 . . . . . . . . 9  |-  0  e.  RR*
76a1i 11 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  0  e.  RR* )
8 xmetcl 21270 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
983expa 1205 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  ( P D x )  e. 
RR* )
1093adantl3 1163 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
11 simpl3 1010 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  R  e.  RR* )
12 xrlelttr 11442 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  ( P D x )  e. 
RR*  /\  R  e.  RR* )  ->  ( (
0  <_  ( P D x )  /\  ( P D x )  <  R )  -> 
0  <  R )
)
137, 10, 11, 12syl3anc 1264 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( (
0  <_  ( P D x )  /\  ( P D x )  <  R )  -> 
0  <  R )
)
145, 13mpand 679 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( ( P D x )  < 
R  ->  0  <  R ) )
1514expimpd 606 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( x  e.  X  /\  ( P D x )  < 
R )  ->  0  <  R ) )
162, 15sylbid 218 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) R )  -> 
0  <  R )
)
1716exlimdv 1768 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( E. x  x  e.  ( P (
ball `  D ) R )  ->  0  <  R ) )
181, 17syl5bi 220 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( P (
ball `  D ) R )  =/=  (/)  ->  0  <  R ) )
19 xblcntr 21350 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P ( ball `  D
) R ) )
20 ne0i 3764 . . . . . 6  |-  ( P  e.  ( P (
ball `  D ) R )  ->  ( P ( ball `  D
) R )  =/=  (/) )
2119, 20syl 17 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  ( P
( ball `  D ) R )  =/=  (/) )
22213expa 1205 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  0  <  R ) )  -> 
( P ( ball `  D ) R )  =/=  (/) )
2322expr 618 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  R  e.  RR* )  ->  (
0  <  R  ->  ( P ( ball `  D
) R )  =/=  (/) ) )
24233impa 1200 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( 0  <  R  ->  ( P ( ball `  D ) R )  =/=  (/) ) )
2518, 24impbid 193 1  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( P (
ball `  D ) R )  =/=  (/)  <->  0  <  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   E.wex 1659    e. wcel 1867    =/= wne 2616   (/)c0 3758   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   0cc0 9528   RR*cxr 9663    < clt 9664    <_ cle 9665   *Metcxmt 18883   ballcbl 18885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-2 10657  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-psmet 18890  df-xmet 18891  df-bl 18893
This theorem is referenced by:  prdsxmslem2  21468  blssioo  21737  metdstri  21792  blbnd  31852  prdsbnd2  31860
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