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Theorem bncmet 18769
Description: The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
iscms.1  |-  X  =  ( Base `  M
)
iscms.2  |-  D  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
Assertion
Ref Expression
bncmet  |-  ( M  e. Ban  ->  D  e.  (
CMet `  X )
)

Proof of Theorem bncmet
StepHypRef Expression
1 bncms 18766 . 2  |-  ( M  e. Ban  ->  M  e. CMetSp )
2 iscms.1 . . 3  |-  X  =  ( Base `  M
)
3 iscms.2 . . 3  |-  D  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
42, 3cmscmet 18768 . 2  |-  ( M  e. CMetSp  ->  D  e.  (
CMet `  X )
)
51, 4syl 15 1  |-  ( M  e. Ban  ->  D  e.  (
CMet `  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    X. cxp 4687    |` cres 4691   ` cfv 5255   Basecbs 13148   distcds 13217   CMetcms 18680  CMetSpccms 18754  Bancbn 18755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-res 4701  df-iota 5219  df-fv 5263  df-cms 18757  df-bn 18758
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