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

Theorem bndmet 26608
Description: A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
bndmet  |-  ( M  e.  ( Bnd `  X
)  ->  M  e.  ( Met `  X ) )

Proof of Theorem bndmet
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbnd 26607 . 2  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. y  e.  RR+  X  =  ( x ( ball `  M
) y ) ) )
21simplbi 446 1  |-  ( M  e.  ( Bnd `  X
)  ->  M  e.  ( Met `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ` cfv 5271  (class class class)co 5874   RR+crp 10370   Metcme 16386   ballcbl 16387   Bndcbnd 26594
This theorem is referenced by:  isbnd3  26611  equivbnd  26617  bnd2lem  26618  equivbnd2  26619  prdsbnd  26620  prdsbnd2  26622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-bnd 26606
  Copyright terms: Public domain W3C validator