Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0totbnd Structured version   Unicode version

Theorem 0totbnd 26483
Description: The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
0totbnd  |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  X
)  <->  M  e.  ( Met `  X ) ) )

Proof of Theorem 0totbnd
Dummy variables  v 
r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5729 . . 3  |-  ( X  =  (/)  ->  ( TotBnd `  X )  =  (
TotBnd `  (/) ) )
21eleq2d 2504 . 2  |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  X
)  <->  M  e.  ( TotBnd `
 (/) ) ) )
3 fveq2 5729 . . . 4  |-  ( X  =  (/)  ->  ( Met `  X )  =  ( Met `  (/) ) )
43eleq2d 2504 . . 3  |-  ( X  =  (/)  ->  ( M  e.  ( Met `  X
)  <->  M  e.  ( Met `  (/) ) ) )
5 0elpw 4370 . . . . . . 7  |-  (/)  e.  ~P (/)
6 0fin 7337 . . . . . . 7  |-  (/)  e.  Fin
7 elin 3531 . . . . . . 7  |-  ( (/)  e.  ( ~P (/)  i^i  Fin ) 
<->  ( (/)  e.  ~P (/) 
/\  (/)  e.  Fin )
)
85, 6, 7mpbir2an 888 . . . . . 6  |-  (/)  e.  ( ~P (/)  i^i  Fin )
9 0iun 4149 . . . . . 6  |-  U_ x  e.  (/)  ( x (
ball `  M )
r )  =  (/)
10 iuneq1 4107 . . . . . . . 8  |-  ( v  =  (/)  ->  U_ x  e.  v  ( x
( ball `  M )
r )  =  U_ x  e.  (/)  ( x ( ball `  M
) r ) )
1110eqeq1d 2445 . . . . . . 7  |-  ( v  =  (/)  ->  ( U_ x  e.  v  (
x ( ball `  M
) r )  =  (/) 
<-> 
U_ x  e.  (/)  ( x ( ball `  M ) r )  =  (/) ) )
1211rspcev 3053 . . . . . 6  |-  ( (
(/)  e.  ( ~P (/) 
i^i  Fin )  /\  U_ x  e.  (/)  ( x ( ball `  M
) r )  =  (/) )  ->  E. v  e.  ( ~P (/)  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) r )  =  (/) )
138, 9, 12mp2an 655 . . . . 5  |-  E. v  e.  ( ~P (/)  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) r )  =  (/)
1413rgenw 2774 . . . 4  |-  A. r  e.  RR+  E. v  e.  ( ~P (/)  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) r )  =  (/)
15 istotbnd3 26481 . . . 4  |-  ( M  e.  ( TotBnd `  (/) )  <->  ( M  e.  ( Met `  (/) )  /\  A. r  e.  RR+  E. v  e.  ( ~P (/)  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) r )  =  (/) ) )
1614, 15mpbiran2 887 . . 3  |-  ( M  e.  ( TotBnd `  (/) )  <->  M  e.  ( Met `  (/) ) )
174, 16syl6rbbr 257 . 2  |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  (/) )  <->  M  e.  ( Met `  X ) ) )
182, 17bitrd 246 1  |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  X
)  <->  M  e.  ( Met `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707    i^i cin 3320   (/)c0 3629   ~Pcpw 3800   U_ciun 4094   ` cfv 5455  (class class class)co 6082   Fincfn 7110   RR+crp 10613   Metcme 16688   ballcbl 16689   TotBndctotbnd 26476
This theorem is referenced by:  prdsbnd2  26505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-en 7111  df-dom 7112  df-fin 7114  df-totbnd 26478
  Copyright terms: Public domain W3C validator