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Theorem bncms 19297
Description: A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bncms  |-  ( W  e. Ban  ->  W  e. CMetSp )

Proof of Theorem bncms
StepHypRef Expression
1 eqid 2436 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
21isbn 19291 . 2  |-  ( W  e. Ban 
<->  ( W  e. NrmVec  /\  W  e. CMetSp  /\  (Scalar `  W
)  e. CMetSp ) )
32simp2bi 973 1  |-  ( W  e. Ban  ->  W  e. CMetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   ` cfv 5454  Scalarcsca 13532  NrmVeccnvc 18629  CMetSpccms 19285  Bancbn 19286
This theorem is referenced by:  bncmet  19300  lssbn  19304  hlcms  19320  sitgclbn  24657
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-bn 19289
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