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Theorem totbndmet 29860
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 29857 . 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 460 1  |-  ( M  e.  ( TotBnd `  X
)  ->  M  e.  ( Met `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810   U.cuni 4240   ` cfv 5581  (class class class)co 6277   Fincfn 7508   RR+crp 11211   Metcme 18170   ballcbl 18171   TotBndctotbnd 29854
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-totbnd 29856
This theorem is referenced by:  totbndss  29865  totbndbnd  29877  prdstotbnd  29882
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