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Theorem totbndmet 26472
Description: The predicate "totally bounded" implies 
M is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
totbndmet  |-  ( M  e.  ( TotBnd `  X
)  ->  M  e.  ( Met `  X ) )

Proof of Theorem totbndmet
Dummy variables  b 
d  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istotbnd 26469 . 2  |-  ( M  e.  ( TotBnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. d  e.  RR+  E. v  e. 
Fin  ( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M ) d ) ) ) )
21simplbi 447 1  |-  ( M  e.  ( TotBnd `  X
)  ->  M  e.  ( Met `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   U.cuni 4007   ` cfv 5446  (class class class)co 6073   Fincfn 7101   RR+crp 10604   Metcme 16679   ballcbl 16680   TotBndctotbnd 26466
This theorem is referenced by:  totbndss  26477  totbndbnd  26489  prdstotbnd  26494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-totbnd 26468
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