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Theorem xbln0 20004
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 3661 . . 3  |-  ( ( P ( ball `  D
) R )  =/=  (/) 
<->  E. x  x  e.  ( P ( ball `  D ) R ) )
2 elbl 19978 . . . . 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 19934 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  0  <_  ( P D x ) )
433expa 1187 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  0  <_  ( P D x ) )
543adantl3 1146 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  0  <_  ( P D x ) )
6 0xr 9445 . . . . . . . . 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 19921 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
983expa 1187 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  ( P D x )  e. 
RR* )
1093adantl3 1146 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
11 simpl3 993 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  R  e.  RR* )
12 xrlelttr 11145 . . . . . . . 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 1218 . . . . . . 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 675 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( ( P D x )  < 
R  ->  0  <  R ) )
1514expimpd 603 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( x  e.  X  /\  ( P D x )  < 
R )  ->  0  <  R ) )
162, 15sylbid 215 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) R )  -> 
0  <  R )
)
1716exlimdv 1690 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( E. x  x  e.  ( P (
ball `  D ) R )  ->  0  <  R ) )
181, 17syl5bi 217 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( P (
ball `  D ) R )  =/=  (/)  ->  0  <  R ) )
19 xblcntr 20001 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P ( ball `  D
) R ) )
20 ne0i 3658 . . . . . 6  |-  ( P  e.  ( P (
ball `  D ) R )  ->  ( P ( ball `  D
) R )  =/=  (/) )
2119, 20syl 16 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  ( P
( ball `  D ) R )  =/=  (/) )
22213expa 1187 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  0  <  R ) )  -> 
( P ( ball `  D ) R )  =/=  (/) )
2322expr 615 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  R  e.  RR* )  ->  (
0  <  R  ->  ( P ( ball `  D
) R )  =/=  (/) ) )
24233impa 1182 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( 0  <  R  ->  ( P ( ball `  D ) R )  =/=  (/) ) )
2518, 24impbid 191 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 184    /\ wa 369    /\ w3a 965   E.wex 1586    e. wcel 1756    =/= wne 2620   (/)c0 3652   class class class wbr 4307   ` cfv 5433  (class class class)co 6106   0cc0 9297   RR*cxr 9432    < clt 9433    <_ cle 9434   *Metcxmt 17816   ballcbl 17818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-po 4656  df-so 4657  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-1st 6592  df-2nd 6593  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-2 10395  df-rp 11007  df-xneg 11104  df-xadd 11105  df-xmul 11106  df-psmet 17824  df-xmet 17825  df-bl 17827
This theorem is referenced by:  prdsxmslem2  20119  blssioo  20387  metdstri  20442  blbnd  28705  prdsbnd2  28713
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