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Theorem totbndmet 31563
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 31560 . 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 1407    e. wcel 1844   A.wral 2756   E.wrex 2757   U.cuni 4193   ` cfv 5571  (class class class)co 6280   Fincfn 7556   RR+crp 11267   Metcme 18726   ballcbl 18727   TotBndctotbnd 31557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-ov 6283  df-totbnd 31559
This theorem is referenced by:  totbndss  31568  totbndbnd  31580  prdstotbnd  31585
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