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Theorem bncms 24049
 Description: A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bncms (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)

Proof of Theorem bncms
StepHypRef Expression
1 eqid 2758 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
21isbn 24043 . 2 (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
32simp2bi 1143 1 (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  ‘cfv 6339  Scalarcsca 16631  NrmVeccnvc 23288  CMetSpccms 24037  Bancbn 24038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-iota 6298  df-fv 6347  df-bn 24041 This theorem is referenced by:  bncmet  24052  lssbn  24057  hlcms  24071  bncssbn  24079  sitgclbn  31833
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