MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xbln0 Structured version   Unicode version

Theorem xbln0 19948
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 3643 . . 3  |-  ( ( P ( ball `  D
) R )  =/=  (/) 
<->  E. x  x  e.  ( P ( ball `  D ) R ) )
2 elbl 19922 . . . . 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 19878 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  0  <_  ( P D x ) )
433expa 1182 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  0  <_  ( P D x ) )
543adantl3 1141 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  0  <_  ( P D x ) )
6 0xr 9426 . . . . . . . . 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 19865 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
983expa 1182 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  ( P D x )  e. 
RR* )
1093adantl3 1141 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
11 simpl3 988 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  R  e.  RR* )
12 xrlelttr 11126 . . . . . . . 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 1213 . . . . . . 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 670 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  X
)  ->  ( ( P D x )  < 
R  ->  0  <  R ) )
1514expimpd 600 . . . . 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 1695 . . 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 19945 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  ( R  e.  RR*  /\  0  <  R ) )  ->  P  e.  ( P ( ball `  D
) R ) )
20 ne0i 3640 . . . . . 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 1182 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  0  <  R ) )  -> 
( P ( ball `  D ) R )  =/=  (/) )
2322expr 612 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  R  e.  RR* )  ->  (
0  <  R  ->  ( P ( ball `  D
) R )  =/=  (/) ) )
24233impa 1177 . 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 960   E.wex 1591    e. wcel 1761    =/= wne 2604   (/)c0 3634   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   0cc0 9278   RR*cxr 9413    < clt 9414    <_ cle 9415   *Metcxmt 17760   ballcbl 17762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-2 10376  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-psmet 17768  df-xmet 17769  df-bl 17771
This theorem is referenced by:  prdsxmslem2  20063  blssioo  20331  metdstri  20386  blbnd  28611  prdsbnd2  28619
  Copyright terms: Public domain W3C validator