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Theorem bncms 22949
 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 2610 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
21isbn 22943 . 2 (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
32simp2bi 1070 1 (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ‘cfv 5804  Scalarcsca 15771  NrmVeccnvc 22196  CMetSpccms 22937  Bancbn 22938 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-bn 22941 This theorem is referenced by:  bncmet  22952  lssbn  22956  hlcms  22970  sitgclbn  29732
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