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Theorem bncms 25379
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 2736 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
21isbn 25373 . 2 (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
32simp2bi 1146 1 (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cfv 6560  Scalarcsca 17301  NrmVeccnvc 24595  CMetSpccms 25367  Bancbn 25368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-bn 25371
This theorem is referenced by:  bncmet  25382  lssbn  25387  hlcms  25401  bncssbn  25409  sitgclbn  34346
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